Question 57
Let $A_{1} A_{2} \ldots A_{n}$ be a convex polygon. Point $P$ inside this polygon is chosen so that its projections $P_{1}, \ldots, P_{n}$ onto lines $A_{1} A_{2}, \ldots, A_{n} A_{1}$ respectively lie on the sides of the polygon. Prove that for arbitrary points $X_{1}, \ldots, X_{n}$ on sides $A_{1} A_{2}, \ldots, A_{n} A_{1}$ respectively,
$$
\max \left\{\frac{X_{1} X_{2}}{P_{1} P_{2}}, \ldots, \frac{X_{n} X_{1}}{P_{n} P_{1}}\right\} \geq 1
$$
$$
\max \left\{\frac{X_{1} X_{2}}{P_{1} P_{2}}, \ldots, \frac{X_{n} X_{1}}{P_{n} P_{1}}\right\} \geq 1
$$