## Related questions with answers

For each of the following statements, prove the statement if it is true or give a counterexample if it is false. a. The midpoint of the common chord of two circles that intersect in two points lies on the line through the centers of the circles. b. If inscribed angle $\angle A B C$ in a circle with center O measures $120^{\circ}$, then central angle $\angle A O C$ measures $120^{\circ}$. c. The angle bisector of any angle inscribed in a circle contains the center of the circle. d. If the line through the midpoints of two chords of a circle contains the center of the circle, then the chords are parallel.

Solution

Verifieda. This is true. The following can be proven through an illustration: (Plotted using a graphing calculator) As shown on the illustration, the midpoint $P$ of the common chord of the two circles coincides with the line that connects the centers of the two circles.

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