MathDB

In triangle

ABC

the bisector of

\angle ACB

intersects

AB

D

. Consider an arbitrary circle

O

passing through

C

and

D

, so that it is not tangent to

BC

CA

. Let O\cap BC \equal{} \{M\} and O\cap CA \equal{} \{N\}. a) Prove that there is a circle

S

so that

DM

and

DN

are tangent to

S

M

and

N

, respectively. b) Circle

S

intersects lines

BC

and

CA

P

and

Q

respectively. Prove that the lengths of

MP

and

NQ

do not depend on the choice of circle

O

Problem Statement