Problem — Congruent chords and the diameter

Text

Two congruent chords $AB$ and $CD$ of a circle with centre $O$ intersect at a point $P$. Consider the diameter through $P$, that is, the line $OP$. Prove that the two chords form congruent angles with this diameter.

Solution

Let $H$ and $K$ be the feet of the perpendiculars drawn from the centre $O$ to the chords $AB$ and $CD$ respectively. Congruent chords are equidistant from the centre, so $\overline{OH} = \overline{OK}$. Consider the triangles $OHP$ and $OKP$: they are right-angled at $H$ and $K$, share the hypotenuse $OP$, and have legs $\overline{OH} = \overline{OK}$. By the congruence criterion for right triangles they are congruent. Consequently $\widehat{OPH} = \widehat{OPK}$: these are precisely the angles formed by the line $OP$ (the diameter) with the two chords $AB$ and $CD$. The chords therefore form congruent angles with the diameter. $\blacksquare$