Derousseau's Generalization of the Malfatti circles

Ajima's example

不朽算法 十四 (Fukyū Sanpō §14)

\(a=252\),  \(b=375\),  \(c=507\).  \(a:b:c=84:125:169\).

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3c (132)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.080485066247&{}:{}&-0.491129575567&{}:{}&1.571614641815&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.112343909928&{}:{}&-0.171340839304&{}:{}&1.058996929376&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.314747575230&{}:{}&-0.879631932355&{}:{}&2.194379507585&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.608695652174&{}:{}&-1.173913043478&{}:{}&1.565217391304&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.443564356436&{}:{}&-1.336633663366&{}:{}&2.780198019802&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{3c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3c}}\) Radical center of the Malfatti circles
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.535937500000&{}:{}&-1.318359375000&{}:{}&1.782421875000&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-11.200000000000&{}:{}&-10.333333333333&{}:{}&22.533333333333&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.311111111111&{}:{}&-0.462962962963&{}:{}&1.774074074074&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.605405405405&{}:{}&-0.729729729730&{}:{}&2.335135135135&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.090964467005&{}:{}&-0.685279187817&{}:{}&1.776243654822&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.324343891403&{}:{}&-1.979185520362&{}:{}&3.303529411765&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.264566929134&{}:{}&-0.244094488189&{}:{}&1.508661417323&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.125526075023&{}:{}&-0.308783165599&{}:{}&1.183257090576&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.357059206246&{}:{}&-0.544567338972&{}:{}&1.187508132726&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.000000000000&{}:{}&4.000000000000&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.280000000000&{}:{}&0.000000000000&{}:{}&0.720000000000&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.076923076923&{}:{}&2.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.421875000000&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-5.333333333333&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.148148148148&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.535937500000&{}:{}&-1.318359375000&{}:{}&1.782421875000&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-11.200000000000&{}:{}&-10.333333333333&{}:{}&22.533333333333&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.311111111111&{}:{}&-0.462962962963&{}:{}&1.774074074074&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.080485066247&{}:{}&-0.491129575567&{}:{}&1.571614641815&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.605405405405&{}:{}&-0.729729729730&{}:{}&2.335135135135&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.090964467005&{}:{}&-0.685279187817&{}:{}&1.776243654822&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.324343891403&{}:{}&-1.979185520362&{}:{}&3.303529411765&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.112343909928&{}:{}&-0.171340839304&{}:{}&1.058996929376&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.264566929134&{}:{}&-0.244094488189&{}:{}&1.508661417323&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.125526075023&{}:{}&-0.308783165599&{}:{}&1.183257090576&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.357059206246&{}:{}&-0.544567338972&{}:{}&1.187508132726&. \end{alignedat} \]
3c (132)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.314747575230&{}:{}&-0.879631932355&{}:{}&2.194379507585&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.535937500000&{}:{}&-1.318359375000&{}:{}&1.782421875000&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-11.200000000000&{}:{}&-10.333333333333&{}:{}&22.533333333333&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.311111111111&{}:{}&-0.462962962963&{}:{}&1.774074074074&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.605405405405&{}:{}&-0.729729729730&{}:{}&2.335135135135&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.090964467005&{}:{}&-0.685279187817&{}:{}&1.776243654822&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.324343891403&{}:{}&-1.979185520362&{}:{}&3.303529411765&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.608695652174&{}:{}&-1.173913043478&{}:{}&1.565217391304&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.000000000000&{}:{}&4.000000000000&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.280000000000&{}:{}&0.000000000000&{}:{}&0.720000000000&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.076923076923&{}:{}&2.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.443564356436&{}:{}&-1.336633663366&{}:{}&2.780198019802&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.605405405405&{}:{}&-0.729729729730&{}:{}&2.335135135135&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.090964467005&{}:{}&-0.685279187817&{}:{}&1.776243654822&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.324343891403&{}:{}&-1.979185520362&{}:{}&3.303529411765&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.000000000000&{}:{}&4.000000000000&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.280000000000&{}:{}&0.000000000000&{}:{}&0.720000000000&,\\C^*_{\mathbf{3c}}&{}\approx{}&-1.076923076923&{}:{}&2.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)