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Irish Mathematical Olympiad

Geometry Difficulty 4.5 AIME Prove it Ireland

Given a point $P$ between the legs of an angle with vertex $A$ . Show, with proof, how to construct a line through $P$ that intersects the legs of the angle at points $B$ and $C$ so that $|PB| = |PC|$ .

Solution

First, draw the circle centre $P$ through $A$ and let $D$ be the second intersection point of this circle with the line $AP$ . Then construct the two lines that pass through $D$ and are parallel to the legs of the given angle. They intersect the legs of the angle at $B$ and $C$ .

By construction, $P$ is the mid-point of $AD$ and $ABDC$ is a parallelogram. Because the diagonals of a parallelogram intersect each other at their mid-points, we obtain $|PB| = |PC|$ .

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