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collinear wanted, tangents to circumcircle related

Source: 2011 XIV All-Ukrainian Tournament of Young Mathematicians, Qualifying p11

May 18, 2021

geometrycollinearUkrainian TYM

Problem Statement

Let

BB_1

and

CC_1

be the altitudes of an acute-angled triangle

ABC

, which intersect its angle bisector

AL

at two different points

P

and

Q

, respectively. Denote by

F

such a point that

PF\parallel AB

and

QF\parallel AC

, and by

T

the intersection point of the tangents drawn at points

B

and

C

to the circumscribed circle of the triangle

ABC

. Prove that the points

A, F

and

T

lie on the same line.