MathDB

The circles

{{w} _ {1}}

and

{{w} _ {2}}

intersect at points

P

and

Q

. Let

AB

and

CD

be parallel diameters of circles

{ {w} _ {1}}

and

{{w} _ {2}}

, respectively. In this case, none of the points

A, B, C, D

coincides with either

P

Q

, and the points lie on the circles in the following order:

A, B, P, Q

on the circle

{{w} _ {1} }

and

C, D, P, Q

on the circle

{{w} _ {2}}

. The lines

AP

and

BQ

intersect at the point

X

, and the lines

CP

and

DQ

intersect at the point

Y, X \ne Y

. Prove that all lines

XY

for different diameters

AB

and

CD

pass through the same point or are all parallel.
(Serdyuk Nazar)

Problem Statement