MathDB

In plane,let a circle

(O)

and two fixed points

B,C

lies in

(O)

such that

BC

not is the diameter.Consider a point

A

varies in

(O)

such that

A\neq B,C

and

AB\neq AC

.Call

D

and

E

respective is intersect of

BC

and internal and external bisector of

\widehat{BAC}

I

is midpoint of

DE

.The line that pass through orthocenter of

\triangle ABC

and perpendicular with

AI

intersects

AD,AE

respective at

M,N

.
1/Prove that

MN

pass through a fixed point
2/Determint the place of

A

such that

S_{AMN}

has maxium value

3