The Circumcircles of Four Triangles

D

H

Let ABC be a triangle. Let DEF be parallel to AC with D on AB and E on BC. Let FGH be parallel to AB with G on BC and H on AC. Let

r

,

r

1

,

r

2

, and

r

3

be the radii of the circumcircles

O

,

O

1

,

O

2

,

O

3

of the triangles ABC, DBE, EFG, and HGC, respectively. Then

r=

r

1

±

r

2

+

r

3

, with the plus sign when F is outside of ABC and minus when F is inside.