Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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2(020)

Malfatti circles

2 (020)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&, \\ P_{\mathbf{2}}&{}\approx{}&0.000462444057&{}:{}&0.997653680320&{}:{}&0.001883875624&, \\ P^-_{\mathbf{2}}&{}\approx{}&-0.005228462733&{}:{}&1.016600415211&{}:{}&-0.011371952478&, \\ P^+_{\mathbf{2}}&{}\approx{}&0.005846253208&{}:{}&0.979729365536&{}:{}&0.014424381255&, \\ Q_{\mathbf{2}}&{}\approx{}&-0.018096102781&{}:{}&1.127668918308&{}:{}&-0.109572815527&, \\ I^\prime_{\mathbf{2}}&{}\approx{}&0.015613713290&{}:{}&0.936247947250&{}:{}&0.048138339461&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{2}}\)
\(P^-_{\mathbf{2}}\)
\(P^+_{\mathbf{2}}\)
\(Q_{\mathbf{2}}\)
\(I^\prime_{\mathbf{2}}\)
2 (020)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2}}&{}\approx{}&-0.297275769760&{}:{}&0.540531570734&{}:{}&0.756744199027&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.005507935911&{}:{}&0.981640213631&{}:{}&0.012851850458&,\\C^\prime_{\mathbf{2}}&{}\approx{}&1.485840203979&{}:{}&2.476400339964&{}:{}&-2.962240543943&, \\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.010975187346&{}:{}&0.987160750894&{}:{}&0.001864061760&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.022197956069&{}:{}&0.887373401289&{}:{}&0.090428642642&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000451606803&{}:{}&0.974273930608&{}:{}&0.025274462589&, \\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.005642409938&{}:{}&1.005606562488&{}:{}&-0.011248972427&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.737068776614&{}:{}&-1.340199855114&{}:{}&1.603131078500&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.005102524720&{}:{}&0.992113554965&{}:{}&0.012988969756&, \\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.107625218511&{}:{}&-0.107625218511&,\\B^*_{\mathbf{2}}&{}\approx{}&0.141742430504&{}:{}&0.000000000000&{}:{}&0.858257569496&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.016309071859&{}:{}&1.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2}}}{B^\prime_{\mathbf{2}}}{C^\prime_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2}}}{B^{\prime\prime}_{\mathbf{2}}}{C^{\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2}}}{B^{\prime\prime\prime}_{\mathbf{2}}}{C^{\prime\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^*_{\mathbf{2}}}{B^*_{\mathbf{2}}}{C^*_{\mathbf{2}}}\)
2 (020)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2}}}}&{}\approx{}&1.621594712201&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2}}}}&{}\approx{}&0.027539679553&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2}}}}&{}\approx{}&7.429201019893&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.200000000000&{}:{}&0.333333333333&{}:{}&0.466666666667&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.297275769760&{}:{}&0.540531570734&{}:{}&0.756744199027&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.005507935911&{}:{}&0.981640213631&{}:{}&0.012851850458&,\\C^\prime_{\mathbf{2}}&{}\approx{}&1.485840203979&{}:{}&2.476400339964&{}:{}&-2.962240543943&. \end{alignedat} \]
2 (020)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2}}&{}\approx{}&0.000462444057&{}:{}&0.997653680320&{}:{}&0.001883875624&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.010975187346&{}:{}&0.987160750894&{}:{}&0.001864061760&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.022197956069&{}:{}&0.887373401289&{}:{}&0.090428642642&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000451606803&{}:{}&0.974273930608&{}:{}&0.025274462589&. \end{alignedat} \]
2 (020)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2}}&{}\approx{}&-0.005228462733&{}:{}&1.016600415211&{}:{}&-0.011371952478&,\\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.005642409938&{}:{}&1.005606562488&{}:{}&-0.011248972427&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.737068776614&{}:{}&-1.340199855114&{}:{}&1.603131078500&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.005102524720&{}:{}&0.992113554965&{}:{}&0.012988969756&. \end{alignedat} \]
2 (020)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2}}&{}\approx{}&0.005846253208&{}:{}&0.979729365536&{}:{}&0.014424381255&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.297275769760&{}:{}&0.540531570734&{}:{}&0.756744199027&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.005507935911&{}:{}&0.981640213631&{}:{}&0.012851850458&,\\C^\prime_{\mathbf{2}}&{}\approx{}&1.485840203979&{}:{}&2.476400339964&{}:{}&-2.962240543943&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.010975187346&{}:{}&0.987160750894&{}:{}&0.001864061760&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.022197956069&{}:{}&0.887373401289&{}:{}&0.090428642642&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000451606803&{}:{}&0.974273930608&{}:{}&0.025274462589&, \end{alignedat} \]
2 (020)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2}}&{}\approx{}&-0.018096102781&{}:{}&1.127668918308&{}:{}&-0.109572815527&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.107625218511&{}:{}&-0.107625218511&,\\B^*_{\mathbf{2}}&{}\approx{}&0.141742430504&{}:{}&0.000000000000&{}:{}&0.858257569496&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.016309071859&{}:{}&1.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2}}&{}\approx{}&0.015613713290&{}:{}&0.936247947250&{}:{}&0.048138339461&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.010975187346&{}:{}&0.987160750894&{}:{}&0.001864061760&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.022197956069&{}:{}&0.887373401289&{}:{}&0.090428642642&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000451606803&{}:{}&0.974273930608&{}:{}&0.025274462589&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.107625218511&{}:{}&-0.107625218511&,\\B^*_{\mathbf{2}}&{}\approx{}&0.141742430504&{}:{}&0.000000000000&{}:{}&0.858257569496&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.016309071859&{}:{}&1.016309071859&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)