MathDB

Let

ABC

be a non-equilateral triangle and let

R

be the radius of its circumcircle. The incircle of

ABC

has

I

as its centre and is tangent to side

CA

in point

D

and to side

CB

in point

E

. Let

A_1

be the point on line

EI

such that

A_1I=R

, with

I

being between

A_1

and

E

. Let

B_1

be the point on line

DI

such that

B_1I=R

, with

I

being between

B_1

and

D

. Let

P

be the intersection of lines

AA_1

and

BB_1

. (a) Prove that

P

belongs to the circumcircle of

ABC

. (b) Let us now also suppose that

AB=1

and

P

coincides with

C

. Determine the possible values of the perimeter of

ABC

Problem Statement