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Triangle sides divided in 3 parts and concurrent lines

Source: Macedonian TST for IMO 2013 - P1 day 1

March 28, 2021

geometrycircumcircle

Problem Statement

The points

A_{1},A_{2},B_{1},B_{2},C_{1},C_{2}

are on the sides

AB

,

BC

and

AC

of an acute triangle

ABC

such that

AA_{1} = A_{1}A_{2} = A_{2}B = \frac{1}{3} AB

,

BB_{1} = B_{1}B_{2} = B_{2}C = \frac{1}{3}BC

and

CC_{1} = C_{1}C_{2} = C_{2}A = \frac{1}{3} AC

. Let

k_{A}, k_{B}

and

k_{C}

be the circumcircles of the triangles

AA_{1}C_{2}

,

BB_{1}A_{2}

and

CC_{1}B_{2}

respectively. Furthermore, let

a_{B}

and

a_{C}

be the tangents to

k_{A}

at

A_{1}

and

C_{2}

,

b_{C}

and

b_{A}

the tangents to

k_{B}

at

B_{1}

and

A_{2}

and

c_{A}

and

c_{B}

the tangents to

k_{C}

at

C_{1}

and

B_{2}

. Show that the perpendicular lines from the intersection points of

a_{B}

and

b_{A}

,

b_{C}

and

c_{B}

,

c_{A}

and

a_{C}

to

AB

,

BC

and

CA

respectively are concurrent.

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