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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6b (321)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.007063953488&{}:{}&1.009556686047&{}:{}&-0.002492732558&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.011819362856&{}:{}&0.953015561455&{}:{}&0.035165075689&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.027087326306&{}:{}&1.069511409919&{}:{}&-0.042424083613&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.317647058824&{}:{}&1.235294117647&{}:{}&0.082352941176&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.078488372093&{}:{}&1.144622093023&{}:{}&-0.066133720930&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{6b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{6b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{6b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{6b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{6b}}\) Radical center of the Malfatti circles
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&5.812500000000&{}:{}&13.671875000000&{}:{}&-18.484375000000&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.020833333333&{}:{}&1.062748015873&{}:{}&-0.041914682540&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.421875000000&{}:{}&0.627790178571&{}:{}&0.794084821429&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.039705882353&{}:{}&1.042279411765&{}:{}&-0.002573529412&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.148170731707&{}:{}&1.200457317073&{}:{}&-0.052286585366&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.007635746606&{}:{}&1.091275452489&{}:{}&-0.083639705882&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.019269983687&{}:{}&0.982998572594&{}:{}&0.036271411093&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.158043806647&{}:{}&0.371742824773&{}:{}&0.470213368580&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.012746710526&{}:{}&1.027789199561&{}:{}&-0.040535910088&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.937500000000&{}:{}&0.062500000000&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.350000000000&{}:{}&0.000000000000&{}:{}&-0.350000000000&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.346153846154&{}:{}&1.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-14.000000000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.031746031746&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.642857142857&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&5.812500000000&{}:{}&13.671875000000&{}:{}&-18.484375000000&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.020833333333&{}:{}&1.062748015873&{}:{}&-0.041914682540&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.421875000000&{}:{}&0.627790178571&{}:{}&0.794084821429&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.007063953488&{}:{}&1.009556686047&{}:{}&-0.002492732558&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.039705882353&{}:{}&1.042279411765&{}:{}&-0.002573529412&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.148170731707&{}:{}&1.200457317073&{}:{}&-0.052286585366&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.007635746606&{}:{}&1.091275452489&{}:{}&-0.083639705882&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.011819362856&{}:{}&0.953015561455&{}:{}&0.035165075689&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.019269983687&{}:{}&0.982998572594&{}:{}&0.036271411093&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.158043806647&{}:{}&0.371742824773&{}:{}&0.470213368580&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.012746710526&{}:{}&1.027789199561&{}:{}&-0.040535910088&. \end{alignedat} \]
6b (321)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.027087326306&{}:{}&1.069511409919&{}:{}&-0.042424083613&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&5.812500000000&{}:{}&13.671875000000&{}:{}&-18.484375000000&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.020833333333&{}:{}&1.062748015873&{}:{}&-0.041914682540&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.421875000000&{}:{}&0.627790178571&{}:{}&0.794084821429&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.039705882353&{}:{}&1.042279411765&{}:{}&-0.002573529412&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.148170731707&{}:{}&1.200457317073&{}:{}&-0.052286585366&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.007635746606&{}:{}&1.091275452489&{}:{}&-0.083639705882&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.317647058824&{}:{}&1.235294117647&{}:{}&0.082352941176&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.937500000000&{}:{}&0.062500000000&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.350000000000&{}:{}&0.000000000000&{}:{}&-0.350000000000&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.346153846154&{}:{}&1.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.078488372093&{}:{}&1.144622093023&{}:{}&-0.066133720930&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.039705882353&{}:{}&1.042279411765&{}:{}&-0.002573529412&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.148170731707&{}:{}&1.200457317073&{}:{}&-0.052286585366&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.007635746606&{}:{}&1.091275452489&{}:{}&-0.083639705882&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.937500000000&{}:{}&0.062500000000&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.350000000000&{}:{}&0.000000000000&{}:{}&-0.350000000000&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.346153846154&{}:{}&1.346153846154&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)