Derousseau's Generalization of the Malfatti circles

Angle Bisectors


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\[\begin{aligned}\overrightarrow{AA^{\prime}} &= \dfrac{1-{\sin\dfrac{A}{2}}-{\sin\dfrac{B}{2}}-{\sin\dfrac{C}{2}}-{\cos\dfrac{A}{2}}+{\cos\dfrac{B}{2}}+{\cos\dfrac{C}{2}}}{2\left(1+{\sin\dfrac{A}{2}}+{\cos\dfrac{B}{2}}+{\cos\dfrac{C}{2}}\right)}\overrightarrow{AI}\\\overrightarrow{BB^{\prime}} &= \dfrac{1+{\sin\dfrac{A}{2}}+{\sin\dfrac{B}{2}}-{\sin\dfrac{C}{2}}-{\cos\dfrac{A}{2}}+{\cos\dfrac{B}{2}}+{\cos\dfrac{C}{2}}}{2\left(1-{\cos\dfrac{A}{2}}-{\sin\dfrac{B}{2}}+{\cos\dfrac{C}{2}}\right)}\overrightarrow{BI}\\\overrightarrow{CC^{\prime}} &= \dfrac{1+{\sin\dfrac{A}{2}}-{\sin\dfrac{B}{2}}+{\sin\dfrac{C}{2}}-{\cos\dfrac{A}{2}}+{\cos\dfrac{B}{2}}+{\cos\dfrac{C}{2}}}{2\left(1-{\cos\dfrac{A}{2}}+{\cos\dfrac{B}{2}}-{\sin\dfrac{C}{2}}\right)}\overrightarrow{CI}\end{aligned}\]

Hiroyasu Kamo