MathDB

Consider a trapezoid

ABCD

with

BC||AD

and

BC<AD

. Let the lines

AB

and

CD

meet at

X

. Let

\omega_1

be the incircle of the triangle

XBC

, and let

\omega_2

be the excircle of the triangle

XAD

which is tangent to the segment

AD

. Denote by

a

and

d

the lines tangent to

\omega_1

, distinct from

AB

and

CD

, and passing through

A

and

D

, respectively. Denote by

b

and

c

the lines tangent to

\omega_2

, distinct from

AB

and

CD

, passing through

B

and

C

respectively. Assume that the lines

a,b,c

and

d

are distinct. Prove that they form a parallelogram.

Problem Statement