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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7b(323)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.002573529412&{}:{}&1.005055147059&{}:{}&-0.002481617647&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.008851156069&{}:{}&0.970691835260&{}:{}&0.020457008671&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.014408682635&{}:{}&1.040653068862&{}:{}&-0.026244386228&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.127272727273&{}:{}&0.818181818182&{}:{}&0.054545454545&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.046052631579&{}:{}&1.110197368421&{}:{}&-0.064144736842&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{7b}}\)
\(P^-_{\mathbf{7b}}\)
\(P^+_{\mathbf{7b}}\)
\(Q_{\mathbf{7b}}\)
\(I^\prime_{\mathbf{7b}}\)
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&3.062500000000&{}:{}&5.859375000000&{}:{}&-7.921875000000&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.012152777778&{}:{}&1.036603009259&{}:{}&-0.024450231481&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.437500000000&{}:{}&0.651041666667&{}:{}&0.786458333333&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.023648648649&{}:{}&1.026182432432&{}:{}&-0.002533783784&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.087500000000&{}:{}&1.171875000000&{}:{}&-0.084375000000&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.002692307692&{}:{}&1.051442307692&{}:{}&-0.048750000000&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.011615044248&{}:{}&0.990735619469&{}:{}&0.020879424779&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.191406250000&{}:{}&0.366210937500&{}:{}&0.442382812500&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.009252265861&{}:{}&1.014680891239&{}:{}&-0.023933157100&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.937500000000&{}:{}&0.062500000000&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.700000000000&{}:{}&0.000000000000&{}:{}&0.300000000000&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.134615384615&{}:{}&0.865384615385&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-6.000000000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.018518518519&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.666666666667&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&3.062500000000&{}:{}&5.859375000000&{}:{}&-7.921875000000&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.012152777778&{}:{}&1.036603009259&{}:{}&-0.024450231481&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.437500000000&{}:{}&0.651041666667&{}:{}&0.786458333333&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.002573529412&{}:{}&1.005055147059&{}:{}&-0.002481617647&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.023648648649&{}:{}&1.026182432432&{}:{}&-0.002533783784&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.087500000000&{}:{}&1.171875000000&{}:{}&-0.084375000000&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.002692307692&{}:{}&1.051442307692&{}:{}&-0.048750000000&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.008851156069&{}:{}&0.970691835260&{}:{}&0.020457008671&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.011615044248&{}:{}&0.990735619469&{}:{}&0.020879424779&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.191406250000&{}:{}&0.366210937500&{}:{}&0.442382812500&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.009252265861&{}:{}&1.014680891239&{}:{}&-0.023933157100&. \end{alignedat} \]
7b (323)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.014408682635&{}:{}&1.040653068862&{}:{}&-0.026244386228&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&3.062500000000&{}:{}&5.859375000000&{}:{}&-7.921875000000&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.012152777778&{}:{}&1.036603009259&{}:{}&-0.024450231481&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.437500000000&{}:{}&0.651041666667&{}:{}&0.786458333333&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.023648648649&{}:{}&1.026182432432&{}:{}&-0.002533783784&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.087500000000&{}:{}&1.171875000000&{}:{}&-0.084375000000&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.002692307692&{}:{}&1.051442307692&{}:{}&-0.048750000000&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.127272727273&{}:{}&0.818181818182&{}:{}&0.054545454545&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.937500000000&{}:{}&0.062500000000&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.700000000000&{}:{}&0.000000000000&{}:{}&0.300000000000&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.134615384615&{}:{}&0.865384615385&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.046052631579&{}:{}&1.110197368421&{}:{}&-0.064144736842&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.023648648649&{}:{}&1.026182432432&{}:{}&-0.002533783784&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.087500000000&{}:{}&1.171875000000&{}:{}&-0.084375000000&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.002692307692&{}:{}&1.051442307692&{}:{}&-0.048750000000&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.937500000000&{}:{}&0.062500000000&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.700000000000&{}:{}&0.000000000000&{}:{}&0.300000000000&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.134615384615&{}:{}&0.865384615385&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)