MathDB

Given an acute and scalene triangle

ABC

with

AB<AC

and random line

(e)

that passes throuh the center of the circumscribed circles

c(O,R)

. Line

(e)

, intersects sides

BC,AC,AB

at points

A_1,B_1,C_1

respectively (point

C_1

lies on the extension of

AB

towards

B

). Perpendicular from

A

on line

(e)

and

AA_1

intersect circumscribed circle

c(O,R)

at points

M

and

A_2

respectively. Prove that a) points

O,A_1,A_2, M

are consyclic b) if

(c_2)

is the circumcircle of triangle

(OBC_1)

and

(c_3)

is the circumcircle of triangle

(OCB_1)

, then circles

(c_1),(c_2)

and

(c_3)

have a common chord

3