Derousseau's Generalization of the Malfatti circles

Angle Bisectors


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\[\begin{aligned}\overrightarrow{AA^{\prime}} &= \dfrac{\sin\dfrac{A}{4}\cos\dfrac{\pi-{B}}{4}\sin\dfrac{\pi-{C}}{4}}{\sqrt{2}\sin\dfrac{\pi-{A}}{4}\cos\dfrac{B}{4}\sin\dfrac{C}{4}}\overrightarrow{AI}\\\overrightarrow{BB^{\prime}} &= \dfrac{\sin\dfrac{\pi-{A}}{4}\cos\dfrac{B}{4}\sin\dfrac{\pi-{C}}{4}}{\sqrt{2}\sin\dfrac{A}{4}\cos\dfrac{\pi-{B}}{4}\sin\dfrac{C}{4}}\overrightarrow{BI}\\\overrightarrow{CC^{\prime}} &= \dfrac{\sin\dfrac{\pi-{A}}{4}\cos\dfrac{\pi-{B}}{4}\sin\dfrac{C}{4}}{\sqrt{2}\sin\dfrac{A}{4}\cos\dfrac{B}{4}\sin\dfrac{\pi-{C}}{4}}\overrightarrow{CI}\end{aligned}\]

Hiroyasu Kamo