Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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2c(130)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&, \\ P_{\mathbf{2c}}&{}\approx{}&1.063690516406&{}:{}&0.661864397016&{}:{}&-0.725554913421&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.912976887089&{}:{}&0.324203415270&{}:{}&-0.237180302359&, \\ P^+_{\mathbf{2c}}&{}\approx{}&1.194102742404&{}:{}&0.954041822919&{}:{}&-1.148144565324&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.125007038184&{}:{}&-0.661448221993&{}:{}&0.536441183809&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&1.321393924846&{}:{}&1.345654435015&{}:{}&-1.667048359862&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{2c}}\)
\(P^-_{\mathbf{2c}}\)
\(P^+_{\mathbf{2c}}\)
\(Q_{\mathbf{2c}}\)
\(I^\prime_{\mathbf{2c}}\)
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.205558966630&{}:{}&0.513897416576&{}:{}&-0.719456383207&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&18.155621565652&{}:{}&25.207495420870&{}:{}&-42.363116986522&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.605717894556&{}:{}&1.009529824259&{}:{}&-0.615247718815&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.172324064859&{}:{}&1.790771526367&{}:{}&-1.963095591226&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.003214693775&{}:{}&0.681089000823&{}:{}&-0.684303694597&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.488616123592&{}:{}&0.926267554174&{}:{}&-1.414883677765&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.728502128525&{}:{}&1.011461601712&{}:{}&-0.739963730237&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.857276127649&{}:{}&0.365433794186&{}:{}&-0.222709921835&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.318698554074&{}:{}&0.468277544578&{}:{}&-0.786976098652&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&5.291287847478&{}:{}&-4.291287847478&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.677124344468&{}:{}&0.000000000000&{}:{}&0.322875655532&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.426891688585&{}:{}&-1.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.102779483315&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&6.051873855217&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.201905964852&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.205558966630&{}:{}&0.513897416576&{}:{}&-0.719456383207&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&18.155621565652&{}:{}&25.207495420870&{}:{}&-42.363116986522&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.605717894556&{}:{}&1.009529824259&{}:{}&-0.615247718815&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&1.063690516406&{}:{}&0.661864397016&{}:{}&-0.725554913421&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.172324064859&{}:{}&1.790771526367&{}:{}&-1.963095591226&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.003214693775&{}:{}&0.681089000823&{}:{}&-0.684303694597&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.488616123592&{}:{}&0.926267554174&{}:{}&-1.414883677765&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.912976887089&{}:{}&0.324203415270&{}:{}&-0.237180302359&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.728502128525&{}:{}&1.011461601712&{}:{}&-0.739963730237&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.857276127649&{}:{}&0.365433794186&{}:{}&-0.222709921835&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.318698554074&{}:{}&0.468277544578&{}:{}&-0.786976098652&. \end{alignedat} \]
2c (130)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&1.194102742404&{}:{}&0.954041822919&{}:{}&-1.148144565324&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.205558966630&{}:{}&0.513897416576&{}:{}&-0.719456383207&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&18.155621565652&{}:{}&25.207495420870&{}:{}&-42.363116986522&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.605717894556&{}:{}&1.009529824259&{}:{}&-0.615247718815&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.172324064859&{}:{}&1.790771526367&{}:{}&-1.963095591226&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.003214693775&{}:{}&0.681089000823&{}:{}&-0.684303694597&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.488616123592&{}:{}&0.926267554174&{}:{}&-1.414883677765&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.125007038184&{}:{}&-0.661448221993&{}:{}&0.536441183809&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&5.291287847478&{}:{}&-4.291287847478&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.677124344468&{}:{}&0.000000000000&{}:{}&0.322875655532&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.426891688585&{}:{}&-1.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&1.321393924846&{}:{}&1.345654435015&{}:{}&-1.667048359862&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.172324064859&{}:{}&1.790771526367&{}:{}&-1.963095591226&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.003214693775&{}:{}&0.681089000823&{}:{}&-0.684303694597&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.488616123592&{}:{}&0.926267554174&{}:{}&-1.414883677765&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&5.291287847478&{}:{}&-4.291287847478&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.677124344468&{}:{}&0.000000000000&{}:{}&0.322875655532&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.426891688585&{}:{}&-1.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)