Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).

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3b (123)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.001690321366&{}:{}&1.312326392984&{}:{}&-0.310636071618&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.109593851268&{}:{}&0.884655678720&{}:{}&0.005750470012&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.178305273327&{}:{}&1.991066602544&{}:{}&-0.812761329217&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.272066863629&{}:{}&1.957340181499&{}:{}&-1.229407045128&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.070106174355&{}:{}&2.521836547934&{}:{}&-1.451730373579&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{3b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3b}}\) Radical center of the Malfatti circles
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&3.077370564310&{}:{}&5.193426410776&{}:{}&-7.270796975086&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.215583153339&{}:{}&1.718610511131&{}:{}&-0.503027357792&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.391815591888&{}:{}&2.319692653146&{}:{}&0.072122938741&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.373338594492&{}:{}&1.799227211857&{}:{}&-0.425888617365&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.011944169131&{}:{}&3.206964507613&{}:{}&-2.195020338482&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002503240932&{}:{}&1.943458331967&{}:{}&-0.940955091035&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.142374411636&{}:{}&1.134996666317&{}:{}&0.007377745320&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.364951664934&{}:{}&0.615899052687&{}:{}&0.019149282379&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.149043216623&{}:{}&1.203096035357&{}:{}&-0.352139251980&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.688901059317&{}:{}&-1.688901059317&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.284190373377&{}:{}&0.000000000000&{}:{}&1.284190373377&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.122035526991&{}:{}&0.877964473009&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-5.193426410776&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.359305255565&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-2.319692653146&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&3.077370564310&{}:{}&5.193426410776&{}:{}&-7.270796975086&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.215583153339&{}:{}&1.718610511131&{}:{}&-0.503027357792&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.391815591888&{}:{}&2.319692653146&{}:{}&0.072122938741&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.001690321366&{}:{}&1.312326392984&{}:{}&-0.310636071618&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.373338594492&{}:{}&1.799227211857&{}:{}&-0.425888617365&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.011944169131&{}:{}&3.206964507613&{}:{}&-2.195020338482&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002503240932&{}:{}&1.943458331967&{}:{}&-0.940955091035&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.109593851268&{}:{}&0.884655678720&{}:{}&0.005750470012&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.142374411636&{}:{}&1.134996666317&{}:{}&0.007377745320&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.364951664934&{}:{}&0.615899052687&{}:{}&0.019149282379&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.149043216623&{}:{}&1.203096035357&{}:{}&-0.352139251980&. \end{alignedat} \]
3b (123)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.178305273327&{}:{}&1.991066602544&{}:{}&-0.812761329217&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&3.077370564310&{}:{}&5.193426410776&{}:{}&-7.270796975086&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.215583153339&{}:{}&1.718610511131&{}:{}&-0.503027357792&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-1.391815591888&{}:{}&2.319692653146&{}:{}&0.072122938741&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.373338594492&{}:{}&1.799227211857&{}:{}&-0.425888617365&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.011944169131&{}:{}&3.206964507613&{}:{}&-2.195020338482&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002503240932&{}:{}&1.943458331967&{}:{}&-0.940955091035&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.272066863629&{}:{}&1.957340181499&{}:{}&-1.229407045128&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.688901059317&{}:{}&-1.688901059317&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.284190373377&{}:{}&0.000000000000&{}:{}&1.284190373377&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.122035526991&{}:{}&0.877964473009&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.070106174355&{}:{}&2.521836547934&{}:{}&-1.451730373579&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.373338594492&{}:{}&1.799227211857&{}:{}&-0.425888617365&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.011944169131&{}:{}&3.206964507613&{}:{}&-2.195020338482&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.002503240932&{}:{}&1.943458331967&{}:{}&-0.940955091035&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.688901059317&{}:{}&-1.688901059317&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.284190373377&{}:{}&0.000000000000&{}:{}&1.284190373377&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.122035526991&{}:{}&0.877964473009&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)