# The Circumcircles of Four Triangles

The Circumcircles of Four Triangles

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 , , , and be the radii of the circumcircles , , , of the triangles ABC, DBE, EFG, and HGC, respectively. Then , with the plus sign when F is outside of ABC and minus when F is inside.

r

r

1

r

2

r

3

O

O

1

O

2

O

3

r=±+

r

1

r

2

r

3