Školjka

%V0 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.