MathDB

In triangle

ABC

, let

A'

is the symmetrical of

A

with respect to the circumcenter

O

ABC

. Prove that: a) The sum of the squares of the tangents segments drawn from

A

and

A'

to the incircle of

ABC

equals

4R^2-4Rr-2r^2

where

R

and

r

are the radii of the circumscribed and inscribed circles of

ABC

respectively. b) The circle with center

A'

and radius

A'I

intersects the circumcircle of

ABC

in a point

L

such that

AL=\sqrt{ AB.AC}

where

I

is the centre of the inscribed circle of

ABC

2