Carnot's Theorem

Given: any triangle ABC

Circumradius: R

Inradius: r

The algebraic sum of the distances from the circumcenter O to the sides: OH+OM+OE

To prove:

OH+OM+OE = R+r

Proof:

1.  By Steiner Step-3

    HN=1/2(r_c-r)

    MD=1/2(r_b-r)

    EF=1/2(r_a-r)

2. ON+OD+OF = 3R

By Steiner:

r_a+ r_b+ r_c - r = 4R

Then: OH+OM+OE = R+r

Q. E. D.