MathDB

VMO 2015 problem 4

Source: VMO 2015

July 31, 2016

geometrycircumcircleangle bisector

Problem Statement

Given a circumcircle

(O)

and two fixed points

B,C

on

(O)

.

BC

is not the diameter of

(O)

. A point

A

varies on

(O)

such that

ABC

is an acute triangle.

E,F

is the foot of the altitude from

B,C

respectively of

ABC

.

(I)

is a variable circumcircle going through

E

and

F

with center

I

.
a) Assume that

(I)

touches

BC

at

D

. Probe that

\frac{DB}{DC}=\sqrt{\frac{\cot B}{\cot C}}

.
b) Assume

(I)

intersects

BC

at

M

and

N

. Let

H

be the orthocenter and

P,Q

be the intersections of

(I)

and

(HBC)

. The circumcircle

(K)

going through

P,Q

and touches

(O)

at

T

(

T

is on the same side with

A

wrt

PQ

). Prove that the interior angle bisector of

\angle{MTN}

passes through a fixed point.

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