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Bosnia and Herzegovina TST 1998 Day 1 Problem 3

Source: Bosnia and Herzegovina Team Selection Test 1998

September 20, 2018

geometryangle bisectororthogonal projection

Problem Statement

Angle bisectors of angles by vertices

A

B

and

C

in triangle

ABC

intersect opposing sides in points

A_1

B_1

and

C_1

, respectively. Let

M

be an arbitrary point on one of the lines

A_1B_1

B_1C_1

and

C_1A_1

. Let

M_1

M_2

and

M_3

be orthogonal projections of point

M

on lines

BC

CA

and

AB

, respectively. Prove that one of the lines

MM_1

MM_2

and

MM_3

is equal to sum of other two