Competition problem — Geometry — Olympiad Maths Prep

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Geometry Difficulty 4.8 AIME Prove it Ukraine

There is isosceles obtuse triangle $ABC$ with vertex in point $B$ given. Perpendicular bisector to side $BC$ intersects lines $AC$ and $AB$ in points $K$ and $M$ respectively. Prove, that point, symmetric to point $A$ with respect to line $BK$ , is on line $CM$ .

(Anton Trigub)

Solution

Let $A_1$ be point, symmetric to $A$ with respect to $BK$ (Fig. 37). At first, $\angle BA_1K = \angle BAK = \angle BCK$ , thus quadrilateral $BA_1CK$ is cyclic. Then,
$\angle A_1CB = \angle A_1KB = \angle AKB = 2\angle BCA = \angle MBC = \angle MCB,$
So we get, that points $C$ , $A_1$ , $M$ are on one line.

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