Question 8
(GBR) IMO2. 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 $AD,BT,CT$ is the sum of the lengths of the other two.
Original formulation. In triangle $ABC$ the angle at $A$ is the smallest. A line through $A$ meets the circumcircle again at the point $U$ lying on the arc $BC$ opposite to $A$. The perpendicular bisectors of $CA$ and $AB$ meet $AU$ at $V$ and $W$, respectively, and the lines $CV,BW$ meet at $T$. Show that $AU=TB+TC$.
(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 $AD,BT,CT$ is the sum of the lengths of the other two.
Original formulation. In triangle $ABC$ the angle at $A$ is the smallest. A line through $A$ meets the circumcircle again at the point $U$ lying on the arc $BC$ opposite to $A$. The perpendicular bisectors of $CA$ and $AB$ meet $AU$ at $V$ and $W$, respectively, and the lines $CV,BW$ meet at $T$. Show that $AU=TB+TC$.