Track / Stage 5 / 31 of 400 #631 of 1964

Problem 631

AIME late

Geometry Difficulty 5.1 Find the answer

17.29*. Inscribe an $n$ -sided polygon in the given circle, one of whose sides passes through the given point, and the other sides are parallel to the given lines.

## §5. Properties of Symmetries and Axes of Symmetry

The source for this one didn't record the answer, so there is nothing to check what you type against. Work it on paper and mark yourself against the solution below.

Official solution

17.29. Under sequential symmetries with respect to lines $l_{1}, \ldots, l_{n-1}$ , which are perpendicular to the given lines and pass through the center of the circle, vertex $A_{1}$ of the desired polygon is transformed into vertex $A_{n}$ . If $n$ is odd, the composition of these symmetries is a rotation by a known angle, so through point $M$ one needs to draw the chord $A_{1} A_{n}$ of known length. If $n$ is even, the considered composition is a symmetry with respect to some line, so from point $M$ one needs to drop a perpendicular to this line.

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