Track / Stage 6 / 101 of 400 #1101 of 1964

Problem 1101

National olympiad, first round

Geometry Difficulty 6.1 Prove it

3. Circles $k, l$ intersect at points $A, B$ . Let $K, L$ be the points of tangency of their common tangent chosen such that point $B$ is an interior point of triangle $AKL$ . On circles $k$ and $l$ , choose points $N$ and $M$ respectively, such that point $A$ is an interior point of segment $MN$ . Prove that quadrilateral $KLMN$ is cyclic if and only if line $MN$ is tangent to the circumcircle of triangle $AKL$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

SOLUTION. From the equality of the arc and the segment angle corresponding to the chord $A K$ of the circle $k$ , it follows (Fig. 1) that $|\angle K N A|=|\angle L K A|$ , and similarly, from the equality of the arc and the segment angle corresponding to the chord $A L$ of the circle $l$ , it follows that $|\angle V L M|=|\angle L A M|$ , where we have denoted by $V$ some point on the half-line opposite to the half-line $L K$ .

Fig. 1

This completes the proof of the statement.

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