MathDB

PM = PN wanted, intersecting circles (2015 Kyiv City MO 10.5)

Source:

September 2, 2020

geometrycirclesequal segments

Problem Statement

Circles

{{w} _ {1}}

and

{{w} _ {2}}

with centers at points

{{O} _ {1}}

and

{{ O} _ {2}}

intersect at points

A

and

B

, respectively. Around the triangle

{{O} _ {1}} {{O} _ {2}} B

circumscribe a circle

w

centered at the point

O

, which intersects the circles

{{w } _ {1}}

and

{{w} _ {2}}

for the second time at points

K

and

L

, respectively. The line

OA

intersects the circles

{{w} _ {1}}

and

{{w} _ {2}}

at the points

M

and

N

, respectively. The lines

MK

and

NL

intersect at the point

P

. Prove that the point

P

lies on the circle

w

and

PM = PN

.
(Vadym Mitrofanov)

Back to Problems View on AoPS