MathDB

Let

ABC

be an acute, non isosceles with

I

is its incenter. Denote

D, E

as tangent points of

(I)

AB,AC

, respectively. The median segments respect to vertex

A

of triangles

ABE

and

ACD

meet

(I)

P,Q,

respectively. Take points

M, N

on the line

DE

such that

AM \perp BE

and

AN \perp C D

respectively. a) Prove that

A

lies on the radical axis of

(MIP)

and

(NIQ)

. b) Suppose that the orthocenter

H

of triangle

ABC

lies on

(I)

. Prove that there exists a line which is tangent to three circles of center

A, B, C

and all pass through

H

Problem Statement