ILL 1992 Problem 27

Let $ABC$ be an arbitrary scalene triangle. Define $\sum$ to be the set of all circles $y$ that have the following properties:(i) $y$ meets each side of $ABC$ in two (possibly coincident) points;(ii) if the points of intersection of $y$ with the sides of the triangle are labeled by $P, Q, R, S, T , U$ , with the points occurring on the sides in orders $\mathcal B(B,P,Q,C), \mathcal B(C, R, S,A), \mathcal B(A, T,U,B)$ , then the following relations of parallelism hold: $TS \parallel BC; PU\parallel CA; RQ\parallel AB$ . (In the limiting cases, some of the conditions of parallelism will hold vacuously; e.g., if $A$ lies on the circle $y$ , then $T$ , $S$ both coincide with $A$ and the relation $TS \parallel BC$ holds vacuously.)

(a) Under what circumstances is $\sum$ nonempty?

(b) Assuming that Σ is nonempty, show how to construct the locus of centers of the circles in the set $\sum$ .

(c) Given that the set $\sum$ has just one element, deduce the size of the largest angle of $ABC.$ (d) Show how to construct the circles in $\sum$ that have, respectively, the largest and the smallest radii.