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ZX = ZY wanted, <BAD=<DAC, 2 circumcircles, 2 angle bisectors, PX//AC, QY//AB

Source: Mathematics Regional Olympiad of Mexico Center Zone 2019 P3

November 14, 2021

geometryequal segmentsequal angles

Problem Statement

Let

ABC

be an acute triangle and

D

a point on the side

BC

such that

\angle BAD = \angle DAC

. The circumcircles of the triangles

ABD

and

ACD

intersect the segments

AC

and

AB

at

E

and

F

, respectively. The internal bisectors of

\angle BDF

and

\angle CDE

intersect the sides

AB

and

AC

at

P

and

Q

, respectively. Points

X

and

Y

are chosen on the side

BC

such that

PX

is parallel to

AC

and

QY

is parallel to

AB

. Finally, let

Z

be the point of intersection of

BE

and

CF

. Prove that

ZX = ZY

.

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