openfreebooks-bookgenerator/content/math/circle-properties/core.html

<div class="book-prose">
  <p>
    Circle theorems connect angles, chords, tangents, and arcs. They are essential for geometry proofs
    and exam reasoning.
  </p>

  <h2 id="key-theorems" class="book-prose__heading">Key circle theorems</h2>
  <ul>
    <li>Angle at center is twice angle at circumference on same arc.</li>
    <li>Angles in the same segment are equal.</li>
    <li>Angle in a semicircle is \(90^\circ\).</li>
    <li>Opposite angles of cyclic quadrilateral sum to \(180^\circ\).</li>
    <li>Tangent is perpendicular to radius at point of contact.</li>
    <li>Two tangents from same external point are equal in length.</li>
    <li>Alternate segment theorem: angle between tangent and chord equals angle in opposite arc.</li>
  </ul>

  <h2 id="proof-usage" class="book-prose__heading">Using theorems in proofs</h2>
  <ol>
    <li>Mark known equal angles/lengths clearly.</li>
    <li>Name theorem used at each step.</li>
    <li>Chain results to target statement.</li>
  </ol>

  <h2 id="cyclic-quadrilateral" class="book-prose__heading">Cyclic quadrilaterals</h2>
  <p>
    A quadrilateral is cyclic if all vertices lie on one circle.
  </p>
  <p>
    Useful tests:
  </p>
  <ul>
    <li>Opposite angles supplementary.</li>
    <li>Exterior angle equals opposite interior angle.</li>
  </ul>

  <h2 id="tangent-chord" class="book-prose__heading">Tangent-chord applications</h2>
  <p>
    When a tangent touches a chord endpoint, use alternate segment theorem to connect tangent angle
    with an inscribed angle subtending that chord.
  </p>

  <h2 id="exam-strategy" class="book-prose__heading">Exam strategy</h2>
  <ul>
    <li>Label all equal angles immediately.</li>
    <li>Write theorem names, not just numbers.</li>
    <li>Avoid assuming symmetry unless given or proved.</li>
    <li>Finish with a clear conclusion sentence.</li>
  </ul>
</div>