<p> Applying Properties of Chords</p><p>Circumcenter—the point that is equidistant from each vertex of the triangle. It is the “center” of the triangle.</p><p> circumcenter of an equilateral triangle </p><p>Chord Bisector Theorem— The perpendicular bisector of a chord passes through the center of the circle (It would be a diameter).</p><p>CD is perpendicular to AB and DE = CE, then AB is a diameter. AB passes through the center of the circle A diameter that is perpendicular to a chord bisects the chord and its corresponding arc.</p><p>If CD is perpendicular to AB and AB is a diameter (which passes through the center point), then CE = DE and m arc CB = m arc DC</p><p>In a circle, or congruent circles, congruent chords are equidistant from the center.</p><p>(converse) In a circle, or congruent circles, chords equidistant from the center are congruent.</p><p> AB is congruent to CD if and only if they are an equal distance away from P EP is congruent to FP AB = 34 CD = 34 Two ways to show two chords are congruent: 1. Show that their corresponding arcs are congruent 2. Show that they are equidistant from the center of the circle. In a circle, a radius perpendicular to a chord bisects the chord. Proof of Theorem 1:</p><p>Statements Reasons 1. 1. Given</p><p>2. 2. Two points determine exactly one line. Perpendicular lines meet to form right 3. 3. angles. A right triangle contains one right 4. 4. angle. 5. 5. Radii in a circle are congruent. Reflexive Property - A segment is 6. 6. congruent to itself. HL - If the hypotenuse and leg of one right triangle are congruent to the 7. 7. corresponding parts of another triangle, the triangles are congruent. CPCTC - Corresponding parts of 8. 8. congruent triangles are congruent. Midpoint of a line segment is the point 9. E is the midpoint of 9. on that line segment that divides the segment two congruent segments. Bisector of a line segment is any line 10. 10. (or subset of a line) that intersects the segment at its midpoint.</p>