Steiner's Theorem

Given: any triangle ABC

Circumradius: R

Inradius: r

Exradius: r_a, r_b, r_c

To prove:

r_a+ r_b+ r_c - r = 4R

Proof:

1. MH + HN = 2R

    H midpoint KG

    H midpoint LQ

2. MH = 1/2(r_a+ r_b)

3. HN = 1/2(r_c - r)

4. In (1):

    r_a+ r_b+ r_c - r = 4R

Q. E. D.