Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.3 AIME, harder Prove it Austria

Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ are externally tangent in $Q$ . The other end-points of the diameter through $Q$ are named $P$ on $k_1$ and $R$ on $k_2$ . We choose two points $A$ and $B$ , one on each of the arcs $PQ$ on $k_1$ . (PBQA is convex.) Furthermore, $C$ is the second common point of the line $AQ$ and $k_2$ , and $D$ is the second common point of $BQ$ with $k_2$ . The lines $PB$ and $RC$ intersect in $U$ and $PA$ and $RD$ intersect in $V$ . Show that a point $Z$ exists, that is common to all possible lines $UV$ .

Solution

A homothety with center $Q$ and ratio $-r_2/r_1$ maps $k_1$ onto $k_2$ .

This homothety maps $A$ to $C$ , $B$ to $D$ , and $P$ to $R$ . It therefore follows that $PB = PU$ and $RD = RV$ are parallel, as are $PA = PV$ and $RC = RU$ . $PURV$ must therefore be a parallelogram (no two of these points can be equal), and the diagonals $PR$ and $UV$ have a common midpoint. It follows that the mid-point $Z$ of $PR$ is also the mid-point of all possible line segments $UV$ , and this is therefore the required common point. qed

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