MathDB

6

Part of 2018 German National Olympiad

Problems(1)

Cevians in a triangle and the foot of a perpendicular

Source: Germany 2018, Problem 6

6/17/2018
Let PP be a point in the interior of a triangle ABCABC and let the rays AP→,BP→\overrightarrow{AP}, \overrightarrow{BP} and CP→\overrightarrow{CP} intersect the sides BC,CABC, CA and ABAB in A1,B1A_1,B_1 and C1C_1, respectively. Let DD be the foot of the perpendicular from A1A_1 to B1C1B_1C_1. Show that CDBD=B1CBC1⋅C1AAB1.\frac{CD}{BD}=\frac{B_1C}{BC_1} \cdot \frac{C_1A}{AB_1}.
geometrycevacevian quotient