Relation between radii of incircle and excircle of a triangle : $r_{1}^{2}=d^{2}+2r_{1}r_{2}$

Question

Problem :

Let $∆ABC$ triangle and $C(D,ED)$ excircle of triangle $ABC$ , $r_{1}=ED$ also $C(G,GF)$ incircle of a triangle $ABC$, $GF=r_2.$

$d=$ distance between centers of the two circles $d=d(D,G)$

Then prove that : $$r_{1}^{2}=d^{2}+2r_{1}r_{2}$$

I know the center of the incircle can be found as the intersection of the three internal angle bisectors and The center of an excircle is the intersection of the internal bisector of one angle

But I don't know which relations I must use to prove the above relations

Answer 1

See Euler's theorem for the triangle.