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Inside the acute triangle $ABC$ the point $P{}$ in on height $AA_1{}$.

Source: Moldova TST 1993

August 8, 2023

geometry

Problem Statement

Inside the acute triangle

ABC

the point

P{}

in on height

AA_1{}

. Lines

BP{}

and

CP{}

intersect the sides

AC{}

and

AB{}

, respectively, in points

B_1{}

and

C_1{}

. Prove that: a)

AA_1{}

is the bisector of the angle

B_1A_1C_1;

b) if the lines

BC

and

B_1C_1

are concurrent, then the position of theri intersection does not depend on

P.