Competition problem — Geometry — Maths Olympiad Prep

Library / /298 of 520

Geometry Difficulty 5.6 AIME, harder Prove it

[ Rotation (other).]

On a plane, there are two identical letters $\Gamma$ . The ends of the short sticks of these letters are denoted by $A$ and $A^{\prime}$ . The long sticks are divided into $n$ equal parts by points $A_{1}, \ldots, A_{n-1} ; A_{1}^{\prime}, \ldots, A_{n-1}^{\prime}$ (the division points are numbered from the ends of the long sticks). The lines $A A_{i}$ and $A^{\prime} A_{i}^{\prime}$ intersect at point $X_{i}$ . Prove that the points $X_{1}, \ldots, X_{n-1}$ form a convex polygon.

Solution

Identical letters $\Gamma$ can be superimposed by a rotation with some center $O$ (if they are superimposed by a parallel translation, then $A A_{\mathrm{i}} \mid A^{\prime} A_{\mathrm{i}}$ '). According to problem 18.25, the point $X_{\mathrm{i}}$ lies on the circumcircle of triangle $A^{\prime} O A$ . It is clear that points lying on the same circle form a convex polygon.

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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.