Derousseau's Generalization of the Malfatti circles

Angle Bisectors


Jump
[Guy]
[Lob & Richmond]
(0**)
(1**)
(2**)
(3**)

\[\begin{aligned}\overrightarrow{AA^{\prime}} &= \dfrac{\cos\dfrac{\alpha}{4}\cos\dfrac{\pi-{\beta}}{4}\cos\dfrac{\pi-{\gamma}}{4}}{\sqrt{2}\cos\dfrac{\pi-{\alpha}}{4}\cos\dfrac{\beta}{4}\cos\dfrac{\gamma}{4}}\overrightarrow{AX}\\\overrightarrow{BB^{\prime}} &= \dfrac{\cos\dfrac{\pi-{\alpha}}{4}\cos\dfrac{\beta}{4}\cos\dfrac{\pi-{\gamma}}{4}}{\sqrt{2}\cos\dfrac{\alpha}{4}\cos\dfrac{\pi-{\beta}}{4}\cos\dfrac{\gamma}{4}}\overrightarrow{BX}\\\overrightarrow{CC^{\prime}} &= \dfrac{\cos\dfrac{\pi-{\alpha}}{4}\cos\dfrac{\pi-{\beta}}{4}\cos\dfrac{\gamma}{4}}{\sqrt{2}\cos\dfrac{\alpha}{4}\cos\dfrac{\beta}{4}\cos\dfrac{\pi-{\gamma}}{4}}\overrightarrow{CX}\end{aligned}\]
Guy L & R \(\alpha\) \(\beta\) \(\gamma\) \(X\)
0 (000) \(A\) \(B\) \(C\) \(I\)
1 (002) \(-A\) \(-B\) \(2\pi-C\) \(I\)
2 (020) \(-A\) \(2\pi-B\) \(-C\) \(I\)
3 (022) \(A\) \(2\pi+B\) \(2\pi+C\) \(I\)
4 (200) \(2\pi-A\) \(-B\) \(-C\) \(I\)
5 (202) \(2\pi+A\) \(B\) \(2\pi+C\) \(I\)
6 (220) \(2\pi+A\) \(2\pi+B\) \(C\) \(I\)
7 (222) \(2\pi-A\) \(2\pi-B\) \(2\pi-C\) \(I\)
0a (011) \(-A\) \(\pi-B\) \(\pi-C\) \(I_A\)
1a (013) \(A\) \(\pi+B\) \(3\pi+C\) \(I_A\)
2a (031) \(A\) \(3\pi+B\) \(\pi+C\) \(I_A\)
3a (033) \(-A\) \(3\pi-B\) \(3\pi-C\) \(I_A\)
4a (211) \(2\pi+A\) \(\pi+B\) \(\pi+C\) \(I_A\)
5a (213) \(2\pi-A\) \(\pi-B\) \(3\pi-C\) \(I_A\)
6a (231) \(2\pi-A\) \(3\pi-B\) \(\pi-C\) \(I_A\)
7a (233) \(2\pi+A\) \(3\pi+B\) \(3\pi+C\) \(I_A\)
0b (101) \(\pi-A\) \(-B\) \(\pi-C\) \(I_B\)
1b (103) \(\pi+A\) \(B\) \(3\pi+C\) \(I_B\)
2b (121) \(\pi+A\) \(2\pi+B\) \(\pi+C\) \(I_B\)
3b (123) \(\pi-A\) \(2\pi-B\) \(3\pi-C\) \(I_B\)
4b (301) \(3\pi+A\) \(B\) \(\pi+C\) \(I_B\)
5b (303) \(3\pi-A\) \(-B\) \(3\pi-C\) \(I_B\)
6b (321) \(3\pi-A\) \(2\pi-B\) \(\pi-C\) \(I_B\)
7b (323) \(3\pi+A\) \(2\pi+B\) \(3\pi+C\) \(I_B\)
0c (110) \(\pi-A\) \(\pi-B\) \(-C\) \(I_C\)
1c (112) \(\pi+A\) \(\pi+B\) \(2\pi+C\) \(I_C\)
2c (130) \(\pi+A\) \(3\pi+B\) \(C\) \(I_C\)
3c (132) \(\pi-A\) \(3\pi-B\) \(2\pi-C\) \(I_C\)
4c (310) \(3\pi+A\) \(\pi+B\) \(C\) \(I_C\)
5c (312) \(3\pi-A\) \(\pi-B\) \(2\pi-C\) \(I_C\)
6c (330) \(3\pi-A\) \(3\pi-B\) \(-C\) \(I_C\)
7c (332) \(3\pi+A\) \(3\pi+B\) \(2\pi+C\) \(I_C\)

Hiroyasu Kamo