MathDB

Two lengths are equal

Source: IMO 2015 Shortlist, G5

July 7, 2016

geometry

Problem Statement

Let

ABC

be a triangle with

CA \neq CB

. Let

D

,

F

, and

G

be the midpoints of the sides

AB

,

AC

, and

H

respectively. A circle

\Gamma

passing through

C

and tangent to

AB

at

D

meets the segments

AF

and

BG

at

H

and

I

, respectively. The points

H'

and

I'

are symmetric to

H

and

I

about

F

and

G

, respectively. The line

H'I'

meets

CD

and

FG

at

Q

and

M

, respectively. The line

CM

meets

\Gamma

again at

P

. Prove that

CQ = QP

.
Proposed by El Salvador

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