MathDB

Given an acute scalene triangle

ABC

inscribed in circle

(O)

. Let

H

be its orthocenter and

M

be the midpoint of

BC

. Let

D

lie on the opposite rays of

HA

so that

BC=2DM

. Let

D'

be the reflection of

D

through line

BC

and

X

be the intersection of

AO

and

MD

.
a) Show that

AM

bisects

D'X

.
b) Similarly, we define the points

E,F

D

and

Y,Z

X

. Let

S

be the intersection of tangent lines from

B,C

with respect to

(O)

. Let

G

be the projection of the midpoint of

AS

to the line

AO

. Show that there exists a point with the same power to all the circles

(BEY),(CFZ),(SGO)

and

(O)

Problem Statement