MathDB

A triangular pyramid

ABCD

is given. A sphere

\omega_A

is tangent to the face

BCD

and to the planes of other faces in points don't lying on faces. Similarly, sphere

\omega_B

is tangent to the face

ACD

and to the planes of other faces in points don't lying on faces. Let

K

be the point where

\omega_A

is tangent to

ACD

, and let

L

be the point where

\omega_B

is tangent to

BCD

. The points

X

and

Y

are chosen on the prolongations of

AK

and

BL

over

K

and

L

such that

\angle CKD = \angle CXD + \angle CBD

and

\angle CLD = \angle CYD +\angle CAD

. Prove that the distances from the points

X

Y

to the midpoint of

CD

are the same.
[hide=thanks ]Thanks to the user Vlados021 for translating the problem.

Problem Statement