Team selection test for 50. IMO

Question

The excircles of ABC touch the sides AB, BC and CA at points M, N and P, respectively. Let I and O be the incenter and the circumcenter of ABC. Prove that if AMNP is a cyclic quadrilateral, then: a) the points M, P and I are collinear; b) the points I, O and N are collinear.

Accepted Answer

By Carnot's theorem, the perpendiculars from the points

M

,

N

and

P

to the lines

A B

,

B C

and

C A

, respectively, have a common point

X

. Then the quadrilateral

A M X P

is cyclic. Now it is easy to see that

X = N

and hence

A N

is a diameter of the circumcircle of

A M N P

.

a) Denote by

I_{B}

and

I_{C}

the respective excenters of

△ A B C

. Then Pappus' theorem for the triples of points

(I_{C}, A, I_{B})

and

(B, N, C)

implies the desired result.

b) Let the incircle of

△ A B C

touches the sides

A B

and

A C

at point

R

and

Q

, respectively. Then the bisectors of

A B

and

A C

coincide with the bisectors of

R M

and

QP

, respectively, and pass through the midpoint of

I N

.