Triangle \(ABC\) is shown together with the angle bisectors of each of its angles. Draw a point \(D\) that is equidistant from sides \(AC\) and \(AB\), but which is closest to side \(BC\).

Priya and Mai are trying to prove that if two chords are congruent, they are equidistant from the center of the circle. Priya draws this picture.

Mai adds the perpendicular segment from the center of the circle to each chord.

Priya says, “I think we should try to use triangles because that is how we proved things congruent before.” Mai says, “I think you’re right, but how? Should we draw in some radii?”

Help them complete the proof.