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<head>
  <meta charset="UTF-8">
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<body>

  <div class="diagram-wrap">
    <svg viewBox="0 0 400 330" aria-label="RHS Proof for Chord Bisector">
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        <path d="M 188,180 L 188,168 L 200,168" fill="none" stroke="#2563eb" stroke-width="2"/>
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        <circle cx="194" cy="174" r="1.5" fill="#2563eb"/>
        <circle cx="206" cy="174" r="1.5" fill="#2563eb"/>
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      <line id="segAM" x1="102" y1="180" x2="200" y2="180" stroke="#64748b" stroke-width="2.5"/>
      <line id="segMB" x1="200" y1="180" x2="298" y2="180" stroke="#64748b" stroke-width="2.5"/>

      
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        <line x1="246" y1="174" x2="246" y2="186" stroke="#16a34a" stroke-width="2.5"/>
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      <line id="sideOM" x1="200" y1="130" x2="200" y2="180" stroke="#0d9488" stroke-width="3"/>
      
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      <line id="hypOA" x1="200" y1="130" x2="102" y2="180" stroke="#ea580c" stroke-width="3"/>
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      <text id="labelS" x="216" y="158" font-size="13" font-weight="700" fill="#0d9488">OM</text>
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  </div>

  <div id="statusBadge" class="status-badge">Right Angle: ∠OMA = ∠OMB = 90°</div>

  <div class="controls">
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    <button class="step-btn" onclick="setStep(1)">2. [H] Hypotenuse</button>
    <button class="step-btn" onclick="setStep(2)">3. [S] Side</button>
    <button class="step-btn" onclick="setStep(3)">4. [∴] AM = MB</button>
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</html>

Perpendicular Bisector of a Chord Theorem and Proof

<!DOCTYPE html>
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<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Perpendicular Bisector of a Chord</title>
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      background: #94a3b8;
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      background: #0284c7;
      transform: scale(1.25);
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</head>
<body>

  <div class="main-grid">
    
    <div class="svg-wrap">
      <svg viewBox="0 0 240 240">
        
        <circle cx="120" cy="110" r="88" fill="none" stroke="#94a3b8" stroke-width="2" stroke-dasharray="4 3" />

        
        <polygon id="triangleOMA" points="120,110 40,144 120,144" fill="#bae6fd" opacity="0.4" />

        
        <line x1="40" y1="144" x2="200" y2="144" stroke="#1e2945" stroke-width="3" stroke-linecap="round" />
        
        
        <line id="lineOM" x1="120" y1="110" x2="120" y2="144" stroke="#0284c7" stroke-width="3" stroke-linecap="round" />

        
        <line id="lineOA" x1="120" y1="110" x2="40" y2="144" stroke="#0d9488" stroke-width="2.5" stroke-dasharray="3 3" />
        
        
        <line x1="120" y1="110" x2="200" y2="144" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="3 3" />

        
        <path d="M 120,135 L 111,135 L 111,144" fill="none" stroke="#0284c7" stroke-width="1.5" />

        
        <circle cx="120" cy="110" r="4.5" fill="#0284c7" />
        <text x="120" y="100" text-anchor="middle" font-size="13" font-weight="700" fill="#1e2945">O</text>

        
        <circle cx="40" cy="144" r="4" fill="#1e2945" />
        <text x="28" y="148" text-anchor="end" font-size="13" font-weight="700" fill="#1e2945">A</text>

        
        <circle cx="200" cy="144" r="4" fill="#1e2945" />
        <text x="212" y="148" text-anchor="start" font-size="13" font-weight="700" fill="#1e2945">B</text>

        
        <circle cx="120" cy="144" r="3.5" fill="#0284c7" />
        <text x="124" y="162" text-anchor="start" font-size="12" font-weight="700" fill="#0284c7">M</text>

        
        
        <text id="labelAM" x="78" y="160" text-anchor="middle" font-size="11" font-weight="600" fill="#1e2945">12 cm</text>
        <text id="labelMB" x="162" y="160" text-anchor="middle" font-size="11" font-weight="600" fill="#64748b">12 cm</text>
        
        
        <text id="labelOA" x="72" y="120" text-anchor="middle" font-size="11" font-weight="700" fill="#0d9488">r = 13</text>

        
        <text id="labelOM" x="132" y="128" text-anchor="start" font-size="11" font-weight="700" fill="#0284c7">d = ?</text>
      </svg>
    </div>

    
    <div class="calc-card">
      <div class="step-title" id="stepInfo">Step 1 of 4: Bisect Chord</div>

      <div class="step-line" id="math0">
        <span>Chord bisected:</span>
        <span class="math-slot" data-math="AM = \frac{AB}{2} = 12\text{ cm}"></span>
      </div>

      <div class="step-line" id="math1">
        <span>Pythagorean theorem:</span>
        <span class="math-slot" data-math="OM^2 + AM^2 = OA^2"></span>
      </div>

      <div class="step-line" id="math2">
        <span>Substitute values:</span>
        <span class="math-slot" data-math="d^2 + 12^2 = 13^2"></span>
      </div>

      <div class="step-line" id="math3">
        <span>Solve for distance:</span>
        <span class="math-slot" data-math="d = \sqrt{169 - 144} = 5\text{ cm}"></span>
      </div>
    </div>
  </div>

  
  <div class="controls">
    <button class="btn" id="prevBtn" onclick="prevStep()">◀ Prev</button>
    <button class="btn" id="playBtn" onclick="togglePlay()">⏸ Pause</button>
    <button class="btn" id="nextBtn" onclick="nextStep()">Next ▶</button>
    <div class="dot-nav">
      <button class="dot-btn" onclick="goToStep(0)" aria-label="Step 1"></button>
      <button class="dot-btn" onclick="goToStep(1)" aria-label="Step 2"></button>
      <button class="dot-btn" onclick="goToStep(2)" aria-label="Step 3"></button>
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      "Step 1 of 4: Bisect Chord",
      "Step 2 of 4: Form Right Triangle",
      "Step 3 of 4: Substitute Values",
      "Step 4 of 4: Solved (d = 5 cm)"
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      dots.forEach((dot, idx) => {
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      });

      // SVG dynamic styles
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      const labelOM = document.getElementById('labelOM');
      const lineOA = document.getElementById('lineOA');
      const lineOM = document.getElementById('lineOM');

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        triangle.setAttribute('opacity', '0.15');
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</html>

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<html lang="en">
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      user-select: none;
      -webkit-user-select: none;
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      height: 100%;
      background: #ffffff;
      font-family: system-ui, -apple-system, sans-serif;
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      justify-content: space-between;
      padding: 12px 16px;
    }
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      display: flex;
      flex-wrap: wrap;
      gap: 8px;
      justify-content: center;
      width: 100%;
      max-width: 380px;
    }
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      background: #f8fafc;
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      color: #475569;
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  <div class="controls">
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      <span class="indicator"></span>Radii OA, OB
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      <span class="indicator"></span>90° at M
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      <button class="step-btn" onclick="setStep(2)">3. Contradict</button>
      <button class="step-btn" onclick="setStep(3)">4. Result</button>
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Radius-Tangent Perpendicularity Theorem and Proof

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
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      <div class="step-pill" onclick="setStep(1)">2. Values</div>
      <div class="step-pill" onclick="setStep(2)">3. Squares</div>
      <div class="step-pill" onclick="setStep(3)">4. Result</div>
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</html>

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<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Tangent Line to a Circle</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      -webkit-user-select: none;
      user-select: none;
    }
    html, body {
      width: 100%;
      height: 100%;
      background-color: #f8fafc;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      overflow: hidden;
    }
    .diagram-container {
      position: relative;
      width: 100%;
      max-width: 360px;
      padding: 8px 12px 28px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
      touch-action: none;
    }
    .label {
      font-size: 12.5px;
      font-weight: 600;
      fill: #1e293b;
      dominant-baseline: central;
    }
    .radius-label {
      fill: #0284c7;
      font-size: 13px;
      font-weight: 700;
      dominant-baseline: central;
    }
    .angle-label {
      fill: #d97706;
      font-size: 11.5px;
      font-weight: 700;
      dominant-baseline: central;
    }
    .hint-pill {
      position: absolute;
      bottom: 4px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      color: #64748b;
      font-size: 11px;
      font-weight: 500;
      padding: 3px 10px;
      border-radius: 9999px;
      pointer-events: none;
      box-shadow: 0 1px 2px rgba(0, 0, 0, 0.04);
      white-space: nowrap;
    }
    .drag-target {
      cursor: grab;
    }
    .drag-target:active {
      cursor: grabbing;
    }
  </style>
</head>
<body>
  <div class="diagram-container">
    <svg id="svg" viewBox="0 0 360 270">
      
      <circle id="circle" cx="180" cy="120" r="75" fill="none" stroke="#1e293b" stroke-width="2.5" />

      
      <line id="tangentLine" x1="65" y1="195" x2="295" y2="195" stroke="#1e293b" stroke-width="2.5" stroke-linecap="round" />

      
      <text id="labelLine" class="label" x="295" y="180" text-anchor="end">Tangent line L</text>

      
      <line id="radiusLine" x1="180" y1="120" x2="180" y2="195" stroke="#0284c7" stroke-width="3" stroke-linecap="round" />

      
      <polyline id="rightAngle" points="180,181 194,181 194,195" fill="none" stroke="#f59e0b" stroke-width="2" stroke-linecap="round" stroke-linejoin="round" />
      <text id="angleText" class="angle-label" x="202" y="174" text-anchor="start">90°</text>

      
      <circle id="centerPt" cx="180" cy="120" r="4.5" fill="#1e293b" />
      <text id="labelO" class="label" x="180" y="104" text-anchor="middle">Center O</text>

      
      <text id="labelRadius" class="radius-label" x="156" y="157" text-anchor="middle">Radius r</text>

      
      <g id="tangentPointGroup" class="drag-target">
        <circle id="touchTarget" cx="180" cy="195" r="22" fill="transparent" />
        <circle id="pointT" cx="180" cy="195" r="5.5" fill="#0284c7" stroke="#ffffff" stroke-width="2" />
      </g>
      <text id="labelT" class="label" x="180" y="217" text-anchor="middle">Point of tangency T</text>
    </svg>
    <div class="hint-pill">Drag point T around circle</div>
  </div>

  <script>
    const svg = document.getElementById('svg');
    const radiusLine = document.getElementById('radiusLine');
    const tangentLine = document.getElementById('tangentLine');
    const rightAngle = document.getElementById('rightAngle');
    const angleText = document.getElementById('angleText');
    const pointT = document.getElementById('pointT');
    const touchTarget = document.getElementById('touchTarget');
    const labelO = document.getElementById('labelO');
    const labelT = document.getElementById('labelT');
    const labelRadius = document.getElementById('labelRadius');
    const labelLine = document.getElementById('labelLine');
    const targetGroup = document.getElementById('tangentPointGroup');

    const ox = 180, oy = 120, r = 75;
    const tangentLen = 115;
    const s = 14; // Right angle square size

    let angle = Math.PI / 2; // Initial position: straight down

    function updateGeometry(rad) {
      const tx = ox + r * Math.cos(rad);
      const ty = oy + r * Math.sin(rad);

      // Radial unit vector (O -> T)
      const ux = Math.cos(rad);
      const uy = Math.sin(rad);

      // Tangent unit vector
      const vx = -Math.sin(rad);
      const vy = Math.cos(rad);

      // Update Radius Line
      radiusLine.setAttribute('x1', ox);
      radiusLine.setAttribute('y1', oy);
      radiusLine.setAttribute('x2', tx);
      radiusLine.setAttribute('y2', ty);

      // Update Tangent Line endpoints
      const lx1 = tx - tangentLen * vx;
      const ly1 = ty - tangentLen * vy;
      const lx2 = tx + tangentLen * vx;
      const ly2 = ty + tangentLen * vy;
      tangentLine.setAttribute('x1', lx1);
      tangentLine.setAttribute('y1', ly1);
      tangentLine.setAttribute('x2', lx2);
      tangentLine.setAttribute('y2', ly2);

      // Right Angle Marker (oriented along +v tangent and -u radius)
      const p1x = tx - s * ux;
      const p1y = ty - s * uy;
      const p2x = tx - s * ux - s * vx;
      const p2y = ty - s * uy - s * vy;
      const p3x = tx - s * vx;
      const p3y = ty - s * vy;
      rightAngle.setAttribute('points', `${p1x},${p1y} ${p2x},${p2y} ${p3x},${p3y}`);

      // 90° label positioned near right angle inside circle
      const ax = tx - (s + 9) * ux - (s + 9) * vx;
      const ay = ty - (s + 9) * uy - (s + 9) * vy;
      angleText.setAttribute('x', ax);
      angleText.setAttribute('y', ay);
      angleText.setAttribute('text-anchor', 'middle');

      // Center O label dynamically placed opposite to radius
      labelO.setAttribute('x', ox - 16 * ux);
      labelO.setAttribute('y', oy - 16 * uy);

      // Tangency Point T & touch target
      pointT.setAttribute('cx', tx);
      pointT.setAttribute('cy', ty);
      touchTarget.setAttribute('cx', tx);
      touchTarget.setAttribute('cy', ty);

      // Label T position (outward along radius)
      labelT.setAttribute('x', tx + 20 * ux);
      labelT.setAttribute('y', ty + 20 * uy);

      // Radius label (offset to the opposite side of 90° marker to avoid crowding)
      const mx = (ox + tx) / 2;
      const my = (oy + ty) / 2;
      const lrx = mx + 20 * vx;
      const lry = my + 20 * vy;
      labelRadius.setAttribute('x', lrx);
      labelRadius.setAttribute('y', lry);

      // Tangent Line label (offset 15px perpendicularly so it never strikes through the line)
      const llx = lx1 - 14 * ux;
      const lly = ly1 - 14 * uy;
      labelLine.setAttribute('x', llx);
      labelLine.setAttribute('y', lly);
      
      // Determine best text anchor based on tangent endpoint location
      if (Math.abs(vx) > 0.3) {
        labelLine.setAttribute('text-anchor', vx < 0 ? 'end' : 'start');
      } else {
        labelLine.setAttribute('text-anchor', 'middle');
      }
    }

    function setAngleFromPointer(clientX, clientY) {
      const rect = svg.getBoundingClientRect();
      const px = ((clientX - rect.left) / rect.width) * 360;
      const py = ((clientY - rect.top) / rect.height) * 270;
      angle = Math.atan2(py - oy, px - ox);
      updateGeometry(angle);
    }

    let isDragging = false;

    function onPointerDown(e) {
      isDragging = true;
      targetGroup.setPointerCapture(e.pointerId);
      setAngleFromPointer(e.clientX, e.clientY);
    }

    function onPointerMove(e) {
      if (!isDragging) return;
      setAngleFromPointer(e.clientX, e.clientY);
    }

    function onPointerUp(e) {
      if (isDragging) {
        try { targetGroup.releasePointerCapture(e.pointerId); } catch (_) {}
        isDragging = false;
      }
    }

    targetGroup.addEventListener('pointerdown', onPointerDown);
    targetGroup.addEventListener('pointermove', onPointerMove);
    targetGroup.addEventListener('pointerup', onPointerUp);
    targetGroup.addEventListener('pointercancel', onPointerUp);

    // Initial positioning
    updateGeometry(angle);
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100vh;
      padding: 12px;
      user-select: none;
    }

    .container {
      width: 100%;
      max-width: 580px;
      display: flex;
      flex-direction: column;
      gap: 12px;
      align-items: center;
    }

    /* Mode selector tabs */
    .controls {
      display: flex;
      justify-content: center;
      gap: 8px;
      flex-wrap: wrap;
    }

    .btn {
      appearance: none;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      color: #475569;
      font-size: 13px;
      font-weight: 600;
      padding: 6px 14px;
      border-radius: 20px;
      cursor: pointer;
      transition: background-color 0.2s ease, color 0.2s ease, border-color 0.2s ease;
    }

    .btn:hover {
      background: #f1f5f9;
    }

    .btn.active[data-view="both"] {
      background: #1e2945;
      color: #ffffff;
      border-color: #1e2945;
    }

    .btn.active[data-view="ab"] {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
    }

    .btn.active[data-view="ba"] {
      background: #d97706;
      color: #ffffff;
      border-color: #d97706;
    }

    .svg-container {
      width: 100%;
      display: flex;
      justify-content: center;
    }

    svg {
      width: 100%;
      height: auto;
      max-height: 380px;
      display: block;
    }

    /* Transition states */
    .ray-ab, .ray-ba, .overlap-zone {
      transition: opacity 0.3s ease;
    }

    .dimmed {
      opacity: 0.15;
    }

    .highlight {
      opacity: 1;
    }
  </style>
</head>
<body>
  <div class="container">
    <div class="controls">
      <button class="btn active" data-view="both">Compare Both</button>
      <button class="btn" data-view="ab">Ray AB</button>
      <button class="btn" data-view="ba">Ray BA</button>
    </div>

    <div class="svg-container">
      <svg viewBox="0 0 540 310" xmlns="http://www.w3.org/2000/svg">
        <defs>
          
          <marker id="arrow-blue" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse">
            <path d="M 0 1.5 L 8 5 L 0 8.5 z" fill="#0284c7" />
          </marker>
          
          <marker id="arrow-amber" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="7" markerHeight="7" orient="auto-start-reverse">
            <path d="M 0 1.5 L 8 5 L 0 8.5 z" fill="#d97706" />
          </marker>
        </defs>

        
        <line x1="170" y1="35" x2="170" y2="265" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="4 4" />
        <line x1="370" y1="35" x2="370" y2="265" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="4 4" />

        
        <g id="group-ab" class="ray-ab">
          
          <line x1="50" y1="70" x2="490" y2="70" stroke="#e2e8f0" stroke-width="3" stroke-linecap="round" />

          
          <line x1="170" y1="70" x2="480" y2="70" stroke="#0284c7" stroke-width="5" stroke-linecap="round" marker-end="url(#arrow-blue)" />

          
          <circle cx="170" cy="70" r="7" fill="#0284c7" />
          <circle cx="170" cy="70" r="3.5" fill="#ffffff" />
          <text x="170" y="44" text-anchor="middle" font-size="16" font-weight="700" fill="#0284c7">A</text>
          <text x="170" y="24" text-anchor="middle" font-size="12" font-weight="600" fill="#0284c7">Endpoint</text>

          
          <circle cx="370" cy="70" r="6" fill="#0284c7" />
          <text x="370" y="44" text-anchor="middle" font-size="16" font-weight="700" fill="#0284c7">B</text>

          
          <rect x="18" y="58" width="90" height="24" rx="12" fill="#e0f2fe" />
          <text x="63" y="75" text-anchor="middle" font-size="14" font-weight="700" fill="#0369a1">Ray AB →</text>
        </g>

        
        <g id="group-center" class="overlap-zone">
          
          <rect x="170" y="112" width="200" height="40" rx="6" fill="#f8fafc" stroke="#94a3b8" stroke-width="1.5" stroke-dasharray="3 3" />
          <text x="270" y="129" text-anchor="middle" font-size="12" font-weight="600" fill="#64748b">Shared Overlap [A, B]</text>
          
          
          <rect x="195" y="136" width="150" height="26" rx="6" fill="#fee2e2" stroke="#fca5a5" stroke-width="1" />
          <text x="270" y="154" text-anchor="middle" font-size="14" font-weight="800" fill="#b91c1c">Ray AB ≠ Ray BA</text>
        </g>

        
        <g id="group-ba" class="ray-ba">
          
          <line x1="50" y1="220" x2="490" y2="220" stroke="#e2e8f0" stroke-width="3" stroke-linecap="round" />

          
          <line x1="370" y1="220" x2="60" y2="220" stroke="#d97706" stroke-width="5" stroke-linecap="round" marker-end="url(#arrow-amber)" />

          
          <circle cx="170" cy="220" r="6" fill="#d97706" />
          <text x="170" y="248" text-anchor="middle" font-size="16" font-weight="700" fill="#d97706">A</text>

          
          <circle cx="370" cy="220" r="7" fill="#d97706" />
          <circle cx="370" cy="220" r="3.5" fill="#ffffff" />
          <text x="370" y="248" text-anchor="middle" font-size="16" font-weight="700" fill="#d97706">B</text>
          <text x="370" y="268" text-anchor="middle" font-size="12" font-weight="600" fill="#d97706">Endpoint</text>

          
          <rect x="432" y="208" width="90" height="24" rx="12" fill="#fef3c7" />
          <text x="477" y="225" text-anchor="middle" font-size="14" font-weight="700" fill="#b45309">← Ray BA</text>
        </g>
      </svg>
    </div>
  </div>

  <script>
    const buttons = document.querySelectorAll('.btn');
    const groupAB = document.getElementById('group-ab');
    const groupBA = document.getElementById('group-ba');
    const groupCenter = document.getElementById('group-center');

    buttons.forEach(btn => {
      btn.addEventListener('click', () => {
        buttons.forEach(b => b.classList.remove('active'));
        btn.classList.add('active');

        const view = btn.dataset.view;

        if (view === 'both') {
          groupAB.classList.remove('dimmed');
          groupBA.classList.remove('dimmed');
          groupCenter.classList.remove('dimmed');
        } else if (view === 'ab') {
          groupAB.classList.remove('dimmed');
          groupBA.classList.add('dimmed');
          groupCenter.classList.add('dimmed');
        } else if (view === 'ba') {
          groupAB.classList.add('dimmed');
          groupBA.classList.remove('dimmed');
          groupCenter.classList.add('dimmed');
        }
      });
    });
  </script>
</body>
</html>

Defining Points, Lines, and Line Segments

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Collinear and Coplanar Points</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      user-select: none;
      -webkit-user-select: none;
    }

    body {
      font-family: system-ui, -apple-system, sans-serif;
      background: #ffffff;
      color: #1e2945;
      padding: 10px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      width: 100%;
      min-height: 100%;
    }

    .panels {
      display: flex;
      flex-wrap: wrap;
      gap: 12px;
      width: 100%;
    }

    .panel {
      flex: 1 1 260px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      position: relative;
    }

    .panel-title {
      font-size: 15px;
      font-weight: 700;
      color: #0f172a;
      margin-bottom: 6px;
      text-align: center;
    }

    .panel-title span.collinear {
      color: #0284c7;
    }

    .panel-title span.non {
      color: #64748b;
    }

    .svg-container {
      width: 100%;
      max-width: 320px;
      height: auto;
      aspect-ratio: 4 / 3;
    }

    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }

    /* Buttons */
    .controls {
      display: flex;
      gap: 6px;
      margin-top: 6px;
      width: 100%;
      justify-content: center;
    }

    .pill-btn {
      font-size: 12px;
      font-weight: 600;
      padding: 4px 10px;
      border-radius: 20px;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      color: #475569;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .pill-btn.active-blue {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.3);
    }

    .pill-btn.active-gray {
      background: #64748b;
      color: #ffffff;
      border-color: #64748b;
      box-shadow: 0 2px 6px rgba(100, 116, 139, 0.3);
    }

    /* SVG Interactive Classes */
    .pt {
      cursor: pointer;
      transition: opacity 0.3s ease, transform 0.3s ease;
    }

    .dimmed {
      opacity: 0.2;
    }

    .highlighted circle {
      stroke-width: 4px;
    }
  </style>
</head>
<body>

  <div class="panels">
    
    <div class="panel">
      <div class="panel-title">
        <span class="collinear">Collinear</span> vs <span class="non">Non-Collinear</span>
      </div>

      <div class="svg-container">
        <svg viewBox="0 0 280 200">
          <defs>
            <marker id="arrow1" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
              <path d="M 0 1.5 L 8 5 L 0 8.5 z" fill="#0284c7" />
            </marker>
            <marker id="arrow1-rev" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="6" markerHeight="6" orient="auto">
              <path d="M 8 1.5 L 0 5 L 8 8.5 z" fill="#0284c7" />
            </marker>
          </defs>

          
          <line id="line1" x1="25" y1="160" x2="255" y2="70" stroke="#0284c7" stroke-width="3" stroke-linecap="round" marker-start="url(#arrow1-rev)" marker-end="url(#arrow1)" />

          
          <g id="pt-S" class="pt" onclick="selectP1('non')">
            <line x1="140" y1="115" x2="140" y2="45" stroke="#94a3b8" stroke-width="1.5" stroke-dasharray="3 3" />
            <circle cx="140" cy="45" r="7" fill="#64748b" stroke="#ffffff" stroke-width="2" />
            <text x="140" y="30" font-size="14" font-weight="700" fill="#475569" text-anchor="middle">S</text>
          </g>

          
          <g id="pt-P" class="pt" onclick="selectP1('col')">
            <circle cx="60" cy="146" r="7" fill="#0284c7" stroke="#ffffff" stroke-width="2" />
            <text x="60" y="172" font-size="14" font-weight="700" fill="#0369a1" text-anchor="middle">P</text>
          </g>

          
          <g id="pt-Q" class="pt" onclick="selectP1('col')">
            <circle cx="140" cy="115" r="7" fill="#0284c7" stroke="#ffffff" stroke-width="2" />
            <text x="140" y="141" font-size="14" font-weight="700" fill="#0369a1" text-anchor="middle">Q</text>
          </g>

          
          <g id="pt-R" class="pt" onclick="selectP1('col')">
            <circle cx="220" cy="84" r="7" fill="#0284c7" stroke="#ffffff" stroke-width="2" />
            <text x="220" y="110" font-size="14" font-weight="700" fill="#0369a1" text-anchor="middle">R</text>
          </g>
        </svg>
      </div>

      <div class="controls">
        <button id="p1-all" class="pill-btn active-blue" onclick="selectP1('all')">All</button>
        <button id="p1-col" class="pill-btn" onclick="selectP1('col')">P, Q, R</button>
        <button id="p1-non" class="pill-btn" onclick="selectP1('non')">S (Off Line)</button>
      </div>
    </div>

    
    <div class="panel">
      <div class="panel-title">
        <span class="collinear">Coplanar</span> vs <span class="non">Non-Coplanar</span>
      </div>

      <div class="svg-container">
        <svg viewBox="0 0 280 200">
          
          <line x1="140" y1="25" x2="140" y2="85" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="3 3" />
          <line x1="50" y1="120" x2="140" y2="85" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="3 3" />
          <line x1="230" y1="120" x2="140" y2="85" stroke="#cbd5e1" stroke-width="1.5" stroke-dasharray="3 3" />

          
          <polygon id="plane-top" points="140,25 230,60 140,95 50,60" fill="rgba(2, 132, 199, 0.16)" stroke="#0284c7" stroke-width="2" stroke-linejoin="round" />

          
          <line x1="50" y1="60" x2="50" y2="120" stroke="#94a3b8" stroke-width="1.5" />
          <line x1="230" y1="60" x2="230" y2="120" stroke="#94a3b8" stroke-width="1.5" />
          <line x1="140" y1="95" x2="140" y2="155" stroke="#94a3b8" stroke-width="1.5" />
          <line x1="50" y1="120" x2="140" y2="155" stroke="#94a3b8" stroke-width="1.5" />
          <line x1="230" y1="120" x2="140" y2="155" stroke="#94a3b8" stroke-width="1.5" />

          
          <g id="pt-A" class="pt" onclick="selectP2('cop')">
            <circle cx="140" cy="45" r="6" fill="#0284c7" stroke="#ffffff" stroke-width="2" />
            <text x="140" y="38" font-size="12" font-weight="700" fill="#0369a1" text-anchor="middle">A</text>
          </g>

          <g id="pt-B" class="pt" onclick="selectP2('cop')">
            <circle cx="185" cy="62" r="6" fill="#0284c7" stroke="#ffffff" stroke-width="2" />
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            <circle cx="140" cy="78" r="6" fill="#0284c7" stroke="#ffffff" stroke-width="2" />
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            <circle cx="140" cy="155" r="7" fill="#64748b" stroke="#ffffff" stroke-width="2" />
            <text x="140" y="177" font-size="14" font-weight="700" fill="#475569" text-anchor="middle">E</text>
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        <button id="p2-non" class="pill-btn" onclick="selectP2('non')">E (Off Plane)</button>
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</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Undefined Geometric Terms</title>
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</head>
<body>

  <div class="grid-container">
    
    <div class="term-card active" onclick="selectCard(0)">
      <span class="badge">0D</span>
      <div class="card-title">Point</div>
      <div class="notation">A</div>
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      <div class="desc">Location</div>
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    <div class="term-card" onclick="selectCard(1)">
      <span class="badge">1D</span>
      <div class="card-title">Line</div>
      <div class="notation"><span class="over-arrow">AB</span></div>
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      <div class="desc">Length</div>
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      <span class="badge">2D</span>
      <div class="card-title">Plane</div>
      <div class="notation">Plane <span class="script-p">P</span></div>
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      <div class="desc">Surface</div>
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</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
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<body>

  
  <div class="nav-bar">
    <button class="tab-btn active" data-step="0">Overview</button>
    <button class="tab-btn" data-step="1">AB ∥ CD</button>
    <button class="tab-btn" data-step="2">AD ∥ BC</button>
    <button class="tab-btn" data-step="3">Shared AC</button>
    <button class="tab-btn" data-step="4">ΔABC ≅ ΔCDA</button>
  </div>

  
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        <path d="M 116 220 A 36 36 0 0 0 111.7 202.9" fill="none" stroke="#2563eb" stroke-width="2.5" stroke-linecap="round"/>
        
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        <text x="62" y="234">A</text>

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        <text x="298" y="234">B</text>

        <circle cx="340" cy="80" r="4.5"/>
        <text x="358" y="76">C</text>

        <circle cx="140" cy="80" r="4.5"/>
        <text x="122" y="76">D</text>
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  </div>

  
  <div id="statusBadge" class="badge-card">
    Parallelogram ABCD with diagonal AC
  </div>

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Properties and Proofs of Parallelograms

Distinguishing Definitions, Postulates, Axioms, and Theorems

Conditional Statements, Converses, Inverses, and Biconditionals

Constructing Direct Synthetic Proofs

Proving Geometric Statements by Contradiction

Disproving Conjectures with Counterexamples

Proving Triangle Properties Using Euclid's Axioms

math

Geometric Reasoning, Axiomatic Systems, and Proof Methods

Angle Pairs Review: Complementary, Supplementary, and Vertical Angles

Angle Properties on Straight Lines

Angle Properties of Parallel Lines and Transversals

Parallel Lines and Transversals: Alternate Interior and Exterior Angles

Using Transversal Angle Relationships to Find Missing Angles

Proving Lines Are Parallel Using Angle Relationships

Constructing the Perpendicular Bisector of a Line Segment

Constructing an Angle Bisector with Compass and Straightedge

Constructing a Perpendicular from a Point to a Line

Constructing a Perpendicular from an External Point

Constructing Parallel Lines Through an External Point

Constructing Regular Polygons: Equilateral Triangle and Square

Using Constructions to Prove Geometric Properties

Lines, Angles, and Geometric Constructions

Congruence of Line Segments and Angles

Proving Triangle Congruence by SSS and SAS

Proving Triangle Congruence by ASA and AAS

Hypotenuse-Leg (HL) Congruence for Right Triangles

Using CPCTC in Multi-Step Geometric Proofs

Isosceles and Equilateral Triangle Theorems and Proofs

Proofs Involving Midpoints and Perpendicular Bisectors

Using Congruent Triangle Proofs in Problems

Triangle Congruence Criteria and Synthetic Proofs

Conditions for Proving a Quadrilateral is a Parallelogram

Special Parallelograms: Rectangles, Rhombi, and Squares

Properties and Proofs of Trapezoids and Kites

Combining Parallel Line Rules with Triangles and Quadrilaterals

Interior and Exterior Angles of Regular Polygons

Regular and Semi-Regular Tessellations of the Plane

Properties and Proofs of Quadrilaterals and Polygons

Similar Polygons: Ratios of Corresponding Sides and Angles

Thales' Parallel Projection and the Triangle Proportionality Theorem

Similar Triangles in Nested and Parallel Line Configurations

Proving Triangle Similarity by AA, SAS, and SSS

Right Triangle Similarity and Geometric Mean Theorems

Proving the Pythagorean Theorem Using Triangle Similarity

Area Ratio Theorem for Similar Triangles

Proving Results Using Area and Volume Ratios of Similar Figures

The Golden Ratio and Golden Rectangles in Geometry

Similarity, Proportions, and Dilations

Equidistant Chords and Chords of Equal Length

Inscribed Angle Theorem and Corollaries

Circle Theorems: Angles in the Same Segment

Angle in a Semicircle: Thales' Circle Theorem

Circle Theorems: Cyclic Quadrilaterals

Converse: If Opposite Angles Are Supplementary, Points Lie on a Circle

Circle Theorems: Chords, Angles, and Cyclic Quadrilaterals

Number of Tangents from a Point on a Circle

Equal Lengths of Tangents from an External Point

The Alternate Segment Theorem and Proof

Intersecting Chords, Secants, and Tangents Segment Lengths

Constructing Tangent Lines to a Circle from an External Point

Common Internal and External Tangents Between Circles

Tangents, Secants, and Circle Power Theorems

定义、命题、定理与推论：数学语言的严格界定与层级关系

命题的结构：区分条件与结论并改写为"如果……那么……"标准形式

原命题与逆命题：互逆命题的构造与真假性独立分析

反例的作用：满足条件但不满足结论的实例与假命题驳斥

定义、命题、定理与反例

证明的意义与必要性：从直觉误区、测量局限到严密逻辑推理

综合法的证明格式："因为……所以……"的规范三段论推导

反证法的含义与逻辑架构：假设不成立、导出矛盾与肯定原结论

反证法的经典应用：证明两直线平行、角不等与无理数命题

数学证明方法：综合法与反证法

OGM.10.GEO.5

OGM.10.GEO.5

Aligned learning resources

Related diagrams