MathDB

The radius of the circle inscribed in triangle

ABC

is equal to

r

, and the radius of the circle tangent to the segment

BC

and the extensions of sides

AB

and

AC

(the exscribed circle corresponding to angle

A

) is equal to

R

. A circle with radius

x < r

is inscribed in angle

\angle BAC

. Tangents to this circles passing through points

B

and

C

and different from

BA

and

AC

intersect at point

A'

. Let

y

be the radius of the circle inscribed in triangle

BCK

. Find the greatest value of the sum

x + y

as x changes from

0

r

. (In this case, it is necessary to prove that this largest value is the same in any triangle with given

r

and

R

Problem Statement