Question 50
Given an acute triangle $ABC$, determine for which points of its interior the following inequalities hold:
$$
1 \leq \frac{\angle APB}{\angle ACB} \leq 2, \quad 1 \leq \frac{\angle BPC}{\angle BAC} \leq 2 \quad \text{and} \quad 1 \leq \frac{\angle CPA}{\angle CBA} \leq 2
$$
$$
1 \leq \frac{\angle APB}{\angle ACB} \leq 2, \quad 1 \leq \frac{\angle BPC}{\angle BAC} \leq 2 \quad \text{and} \quad 1 \leq \frac{\angle CPA}{\angle CBA} \leq 2
$$