Question 57
Let $ABC$ be a triangle each of whose angles is greater than $30^\circ$. Suppose a circle, with centre $T$, cuts the segments $BC$ in $P, Q$; $CA$ in $K, L$; and $AB$ in $M, N$, such that $P, Q, K, L, M, N$ are on the circle in counter-clockwise direction in that order. Suppose further that the triangles $TQK$, $TLM$ and $TNP$ are all equilateral. Prove that:
(i) the radius of the circle is $\frac{2abc}{a^2 + b^2 + c^2 + 4\sqrt{3}\Delta}$;
(ii) $a \cdot AT = b \cdot BT = c \cdot CT$.
(i) the radius of the circle is $\frac{2abc}{a^2 + b^2 + c^2 + 4\sqrt{3}\Delta}$;
(ii) $a \cdot AT = b \cdot BT = c \cdot CT$.