MathDB

Given is an acute angled triangle

\triangle ABC

with side lengths

a

b

and

c

(in an usual way) and circumcenter

O

. Angle bisector of angle

\angle BAC

intersects circumcircle at points

A

and

A_{1}

. Let

D

be projection of point

A_{1}

onto line

AB

L

and

M

be midpoints of

AC

and

AB

, respectively. (i) Prove that AD\equal{}\frac{1}{2}(b\plus{}c) (ii) If triangle

\triangle ABC

is an acute angled prove that A_{1}D\equal{}OM\plus{}OL

Problem Statement