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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>

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

math

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

圆的定义：平面内到定点距离等于定长的点的集合

圆的相关概念：弦、直径、弧（优弧与劣弧）、半圆与等圆

圆的对称性：无数条对称轴与绕圆心的旋转不变性

圆的画法与车轮为什么是圆形的数学解析

圆的基础概念与几何探究

点与圆的位置关系：点在圆内、圆上与圆外的距离判定

圆心角、弧、弦之间的关系：圆的旋转不变性与等对等定理

垂径定理的发现与证明：垂直于弦的直径平分弦及所对两条弧

垂径定理中的直角三角形：弦长、弦心距与半径的勾股计算

垂径定理的实际应用：赵州桥拱高与圆形管道水深模型

圆的对称性与垂径定理

认识圆周角：顶点在圆上且两边都与圆相交的角的特征

圆周角定理的证明：一条弧所对的圆周角等于它所对圆心角的一半

圆周角定理推论（一）：同弧或等弧所对的圆周角相等

圆周角定理推论（二）：直径所对圆周角是直角与90度角所对弦是直径

圆内接四边形的性质：对角互补与外角等于内对角

圆周角定理与圆内接四边形

直线与圆的位置关系：相交、相切、相离与圆心距d与半径r的数量关系

切线的判定定理：经过半径外端并且垂直于这条半径的直线是切线

切线的性质定理：圆的切线垂直于经过切点的半径

切线长定理：过圆外一点所画的两条切线长相等且平分夹角

过圆外一点作圆的切线的尺规作图与切线综合计算

直线与圆的位置关系及切线性质与判定

确定圆的条件与外接圆：过不在同一直线上的三点作圆

三角形的外心：三边垂直平分线的交点及在不同三角形中的位置

OGM.10.GEO.4

Aligned learning resources

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