The Radii of Four Incircles
The Radii of Four Incircles
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 incircles O, , , of the triangles ABC, DBE, FEG, and CGH, 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
1
O
2
O
3
r=±+
r
1
r
2
r
3