MathDB

In the scalene acute triangle

ABC

O

is the circumcenter.

AD, BE, CF

are three altitudes. And

H

is the orthocenter. Let

G

be the reflection point of

O

through

BC

. Draw the diameter

EK

\odot (GHE)

, and the diameter

FL

\odot (GHF)

.
a) If

AK, AL

and

DE, DF

intersect at

U, V

respectively, prove that

UV\parallel EF

.
b) Suppose

S

is the intersection of the two tangents of the circumscribed circle of

\triangle ABC

B

and

C

T

is the intersection of

DS

and

HG

. And

M,N

are the projection of

H

TE,TF

respectively. Prove that

M,N,E,F

are concyclic.

Problem Statement