Given a diameter of a circle bisecting the angle formed by two intersecting chords, Prove the chords are equal

Question

In a circle, a diameter bisects the angle formed by two intersecting chords.

Prove that the chords are equal

Answer 1

Triangles IEB IEC are congruent (sides and included angle common)

The three lines are $ concurrent $ at E. $Three $ vertically opposite angles are same. Mark them separately as $ p,q,r $. Choose from among them conveniently .

BE = EC

Triangles AEI DEI are congruent (sides and included angle common)

DE = AE

Total chord length is same;

AE + EC = DE + EB.

Answer 2

Angle AEB= angle CED (vertical angles are equal)

Arc AC = arc BD (arc addition postulate)

AC=BD (equal arcs have equal chords)