IMO 1997 Problem 2

It is known that $\angle BAC$ is the smallest angle in the triangle $ABC$ . The points $B$ and $C$ divide the circumcircle of the triangle into two arcs. Let $U$ be an interior point of the arc between $B$ and $C$ which does not contain $A$ . The perpendicular bisectors of $AB$ and $AC$ meet the line $AU$ at $V$ and $W$ , respectively. The lines $BV$ and $CW$ meet at $T$ .

Show that $AU = TB + TC$ .

Alternative formulation:

Four different points $A,B,C,D$ are chosen on a circle $\Gamma$ such that the triangle $BCD$ is not right-angled. Prove that:

(a) The perpendicular bisectors of $AB$ and $AC$ meet the line $AD$ at certain points $W$ and $V,$ respectively, and that the lines $CV$ and $BW$ meet at a certain point $T.$

(b) The length of one of the line segments $AD, BT,$ and $CT$ is the sum of the lengths of the other two.