Derousseau's Generalization of the Malfatti circles

Angle Bisectors (3)


[Other solutions]
[Guy]
[Lob & Richmond]

\(\mathbf{4b}\) \((301)\)

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

Hiroyasu Kamo