Problem 299: Intersecting Circles, Common Chord, Secant, Midpoint, Congruence. Level: High School, College, SAT Prep.

In the figure, the congruent circles O and Q intersect at A and B. C is the midpoint of chord AB. Lines DA and DB intersect circle Q at E and F, respectively. Lines EC and FC intersect circle O at G and H, respectively. Prove that chords DG and DH are congruent.