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midpoints, parallels, intersections, collinear in the end

Source: Vietnamese MO (VMO) 1972

August 24, 2018

geometryParallel Linesparallelcollinear

Problem Statement

ABC

is a triangle.

U

is a point on the line

BC

.

I

is the midpoint of

BC

. The line through

C

parallel to

AI

meets the line

AU

at

E

. The line through

E

parallel to

BC

meets the line

AB

at

F

. The line through

E

parallel to

AB

meets the line

BC

at

H

. The line through

H

parallel to

AU

meets the line

AB

at

K

. The lines

HK

and

FG

meet at

T. V

is the point on the line

AU

such that

A

is the midpoint of

UV

. Show that

V, T

and

I

are collinear.

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