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Circumcircles intersect B_1C_1 at isotomic points

Source: Sharygin 2019 Finals Day 1 Grade 10 P2

July 30, 2019

geometrycircumcircle

Problem Statement

Let

A_1

,

B_1

,

C_1

be the midpoints of sides

BC

,

AC

and

AB

of triangle

ABC

,

AK

be the altitude from

A

, and

L

be the tangency point of the incircle

\gamma

with

BC

. Let the circumcircles of triangles

LKB_1

and

A_1LC_1

meet

B_1C_1

for the second time at points

X

and

Y

respectively, and

\gamma

meet this line at points

Z

and

T

. Prove that

XZ = YT

.

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