MathDB

The incircle

\mathcal{C}_1

of triangle

ABC

touches the sides

AB

and

AC

at the points

D

and

E

, respectively. The incircle

\mathcal{C}_2

of the triangle

ADE

touches the sides

AB

and

AC

at the points

P

and

Q

, and intersects the circle

\mathcal{C}_1

at the points

M

and

n

. Prove that
(a) the center of the circle

\mathcal{C}_2

lies on the circle

\mathcal{C}_1

.
(b) the four points

M,N,P,Q

in appropriate order form a rectangle if and only if twice the radius of

\mathcal{C}_1

is three times the radius of

\mathcal{C}_2

Problem Statement