Track / Stage 5 / 212 of 400 #812 of 1964

Problem 812

AIME late

Geometry Difficulty 5.6 Find the answer

40. Pentagon $A B C D E$ is inscribed in a circle. Points $M, Q, N$ and $P$ are the feet of the perpendiculars dropped from vertex $E$ to sides $A B, B C, C D$ (or their extensions) and diagonal $A D$ , respectively. It is known that $|E P|=d$ , and the ratio of the area of triangle $M Q E$ to the area of triangle $P N E$ is $k$ . Find $|E M|$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

40. Points $E, M, B$ and $Q$ (Fig. 7) lie on the same circle

Fig. 7. with diameter $B E$ , and points $E, P$ , $D$ and $N$ lie on the circle with diameter $E D$ .

Thus, $\widehat{E M Q}=\widehat{E B Q}=$ $=180^{\circ}-\widehat{E D C}=\widehat{E D N}=\widehat{E P N}$ ; similarly, $\widehat{E Q N}=\widehat{E N P}$ , i.e., $\triangle E M Q$ is similar to $\triangle E P N$ with a similarity coefficient of $\sqrt{k}$ . (For completeness, other cases of point placement must also be considered.)

Answer: $d V \bar{k}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.