MathDB

Three circles are constructed for the triangle

ABC

: the circle

{{w} _ {A}}

passes through the vertices

B

and

C

and intersects the sides

AB

and

AC

at points

{{A} _ {1}}

and

{{A} _ {2}}

respectively, the circle

{{w} _ {B}}

passes through the vertices

A

and

C

and intersects the sides

BA

and

BC

at the points

{{B} _ {1}}

and

{{B} _ {2}}

{{w} _ {C}}

passes through the vertices

A

and

B

and intersects the sides

CA

and

CB

at the points

{{C} _ {1}}

and

{{C} _ {2}}

. Let

{{A} _ {1}} {{A} _ {2}} \cap {{B} _ {1}} {{B} _ {2}} = {C} '

{{A} _ {1}} {{A} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {B} '

{ {B} _ {1}} {{B} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {A} '

is Prove that the perpendiculars, which are omitted from the points

{A} ', \, \, {B}', \, \, {C} '

to the lines

BC

CA

and

AB

respectively intersect at one point.
(Rudenko Alexander)

Problem Statement