Question 21
Point $P$ inside an acute-angled triangle $A_1A_2A_3$ is chosen so that its projections $P_1, P_2, P_3$ onto the sides $A_1A_2$, $A_2A_3$, $A_3A_1$ respectively lie on the sides of the triangle.
Prove that for points $X_1$, $X_2$, $X_3$ on the sides $A_1A_2$, $A_2A_3$, $A_3A_1$ respectively, $$\max \left\{ \frac{X_1X_2}{P_1P_2}, \frac{X_2X_3}{P_2P_3}, \frac{X_3X_1}{P_3P_1} \right\} \ge 1$$ if
a) $X_1$, $X_2$, $X_3$ are the midpoints of the corresponding sides;
b) $X_1$, $X_2$, $X_3$ are the feet of the corresponding altitudes;
c) $X_1$, $X_2$, $X_3$ are arbitrary points on the corresponding sides.
(IMO-2010 Shortlist, Problem G 3, modified)
Prove that for points $X_1$, $X_2$, $X_3$ on the sides $A_1A_2$, $A_2A_3$, $A_3A_1$ respectively, $$\max \left\{ \frac{X_1X_2}{P_1P_2}, \frac{X_2X_3}{P_2P_3}, \frac{X_3X_1}{P_3P_1} \right\} \ge 1$$ if
a) $X_1$, $X_2$, $X_3$ are the midpoints of the corresponding sides;
b) $X_1$, $X_2$, $X_3$ are the feet of the corresponding altitudes;
c) $X_1$, $X_2$, $X_3$ are arbitrary points on the corresponding sides.
(IMO-2010 Shortlist, Problem G 3, modified)