Derousseau's Generalization of the Malfatti circles

Angle Bisectors


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\(\mathbf{7a}\)   \((233)\)

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

Hiroyasu Kamo