MathDB

KL=BC /2 wanted, circle passing through A tangent to BC at D, angle bisector

Source: Champions Tournament (Ukraine) - Турнір чемпіонів - 2017 Seniors p4

September 2, 2020

equal segmentsgeometrycircleangle bisectorChampions Tournament

Problem Statement

Let

AD

be the bisector of triangle

ABC

. Circle

\omega

passes through the vertex

A

and touches the side

BC

at point

D

. This circle intersects the sides

AC

and

AB

for the second time at points

M

and

N

respectively. Lines

BM

and

CN

intersect the circle for the second time

\omega

at points

P

and

Q

, respectively. Lines

AP

and

AQ

intersect side

BC

at points

K

and

L

, respectively. Prove that

KL=\frac12 BC