MathDB

collinearity wanted, median, altitudes, 3 circles related

Source: Balkan MO Shortlist 2008 G6

April 6, 2020

geometrycollinearcirclestangentaltitudesmedian

Problem Statement

On triangle

ABC

the

AM

(

M\in BC

) is median and

BB_1

and

CC_1

(

B_1 \in AC,C_1 \in AB

) are altitudes. The stright line

d

is perpendicular to

AM

at the point

A

and intersect the lines

BB_1

and

CC_1

at the points

E

and

F

respectively. Let denoted with

\omega

the circle passing through the points

E, M

and

F

and with

\omega_1

and with

\omega_2

the circles that are tangent to segment

EF

and with

\omega

at the arc

EF

which is not contain the point

M

. If the points

P

and

Q

are intersections points for

\omega_1

and

\omega_2

then prove that the points

P, Q

and

M

are collinear.

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