Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.1 AIME, harder Prove it

Geometry

Let $A B$ and $C D$ be two diameters of the circle $\mathcal{C}$ . For an arbitrary point $P$ on $\mathcal{C}$ , let $R$ and $S$ be the feet of the perpendiculars from $P$ to $A B$ and $C D$ , respectively. Show that the length of $R S$ is independent of the choice of $P$ .

Solution

Let $O$ be the centre of $\mathcal{C}$ . Then $P, R, S$ , and $O$ are points on a circle $\mathcal{C}^{\prime}$ with diameter $O P$ , equal to the radius of $\mathcal{C}$ . The segment $R S$ is a chord in this circle subtending the angle $B O D$ or a supplementary angle. Since the angle as well as radius of $\mathcal{C}^{\prime}$ are independent of $P$ , so is $R S$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.