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BAD CAT geometry

Source: 2019 Philippine IMO TST1 Problem 6

May 4, 2022

geometryanglesparallelogramisogonal lines

Problem Statement

Let

D

be an interior point of triangle

ABC

. Lines

BD

and

CD

intersect sides

AC

and

AB

at points

E

and

F

, respectively. Points

X

and

Y

are on the plane such that

BFEX

and

CEFY

are parallelograms. Suppose lines

EY

and

FX

intersect at a point

T

inside triangle

ABC

. Prove that points

B

,

C

,

E

, and

F

are concyclic if and only if

\angle BAD = \angle CAT

.

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