Question 29
Let $AB\Gamma$ be an acute angled triangle with $AB \le A\Gamma$ and $c(O, R)$ its circumcircle. The perpendicular from $A$ to the tangent of $c(O, R)$ at $\Gamma$ intersects it at $\Delta$.
(α) If the triangle $AB\Gamma$ is isosceles with $AB = A\Gamma$, prove that: $\Gamma\Delta = \frac{B\Gamma}{2}$.
(β) If $\Gamma\Delta = \frac{B\Gamma}{2}$, prove that the triangle $AB\Gamma$ is isosceles.
(α) If the triangle $AB\Gamma$ is isosceles with $AB = A\Gamma$, prove that: $\Gamma\Delta = \frac{B\Gamma}{2}$.
(β) If $\Gamma\Delta = \frac{B\Gamma}{2}$, prove that the triangle $AB\Gamma$ is isosceles.