Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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1b(103)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&, \\ P_{\mathbf{1b}}&{}\approx{}&0.897290830879&{}:{}&-0.414625789317&{}:{}&0.517334958438&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.907562864444&{}:{}&-0.160003302040&{}:{}&0.252440437596&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.889516139794&{}:{}&-0.607344316691&{}:{}&0.717828176896&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.100542534547&{}:{}&0.273172980514&{}:{}&-0.373715515061&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.853783559874&{}:{}&-0.899114392204&{}:{}&1.045330832330&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.971350761400&{}:{}&-0.343790863202&{}:{}&0.372440101802&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.415884804830&{}:{}&-0.497185297387&{}:{}&1.081300492557&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&2.056997126177&{}:{}&-4.936793102824&{}:{}&3.879795976647&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.691884103021&{}:{}&-1.243830498087&{}:{}&1.551946395066&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.042031448097&{}:{}&-0.642816920700&{}:{}&0.600785472603&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.829043531200&{}:{}&-0.383089647940&{}:{}&0.554046116741&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.591485842436&{}:{}&-0.707114232250&{}:{}&1.115628389814&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.081991937685&{}:{}&-0.382950173116&{}:{}&0.300958235430&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.828414643244&{}:{}&-0.146049473343&{}:{}&0.317634830099&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-2.716989200105&{}:{}&3.716989200105&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.514173943788&{}:{}&0.000000000000&{}:{}&-0.514173943788&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.801142975005&{}:{}&0.198857024995&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.171895431601&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.499061765796&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&2.468396551412&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.971350761400&{}:{}&-0.343790863202&{}:{}&0.372440101802&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.415884804830&{}:{}&-0.497185297387&{}:{}&1.081300492557&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&2.056997126177&{}:{}&-4.936793102824&{}:{}&3.879795976647&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.897290830879&{}:{}&-0.414625789317&{}:{}&0.517334958438&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.691884103021&{}:{}&-1.243830498087&{}:{}&1.551946395066&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.042031448097&{}:{}&-0.642816920700&{}:{}&0.600785472603&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.829043531200&{}:{}&-0.383089647940&{}:{}&0.554046116741&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.907562864444&{}:{}&-0.160003302040&{}:{}&0.252440437596&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.591485842436&{}:{}&-0.707114232250&{}:{}&1.115628389814&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.081991937685&{}:{}&-0.382950173116&{}:{}&0.300958235430&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.828414643244&{}:{}&-0.146049473343&{}:{}&0.317634830099&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.889516139794&{}:{}&-0.607344316691&{}:{}&0.717828176896&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.971350761400&{}:{}&-0.343790863202&{}:{}&0.372440101802&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.415884804830&{}:{}&-0.497185297387&{}:{}&1.081300492557&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&2.056997126177&{}:{}&-4.936793102824&{}:{}&3.879795976647&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.691884103021&{}:{}&-1.243830498087&{}:{}&1.551946395066&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.042031448097&{}:{}&-0.642816920700&{}:{}&0.600785472603&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.829043531200&{}:{}&-0.383089647940&{}:{}&0.554046116741&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.100542534547&{}:{}&0.273172980514&{}:{}&-0.373715515061&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-2.716989200105&{}:{}&3.716989200105&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.514173943788&{}:{}&0.000000000000&{}:{}&-0.514173943788&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.801142975005&{}:{}&0.198857024995&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.853783559874&{}:{}&-0.899114392204&{}:{}&1.045330832330&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.691884103021&{}:{}&-1.243830498087&{}:{}&1.551946395066&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.042031448097&{}:{}&-0.642816920700&{}:{}&0.600785472603&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.829043531200&{}:{}&-0.383089647940&{}:{}&0.554046116741&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-2.716989200105&{}:{}&3.716989200105&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.514173943788&{}:{}&0.000000000000&{}:{}&-0.514173943788&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.801142975005&{}:{}&0.198857024995&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)