Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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3a(033)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.001907932339&{}:{}&0.027605922934&{}:{}&0.974302009405&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.020353108133&{}:{}&-0.007855173601&{}:{}&0.987502065469&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.021532679463&{}:{}&0.058867491912&{}:{}&0.962665187551&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.018224037448&{}:{}&0.268478726828&{}:{}&0.713297235724&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.026021777990&{}:{}&0.127785328297&{}:{}&0.898236449692&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{3a}}\)
\(P^-_{\mathbf{3a}}\)
\(P^+_{\mathbf{3a}}\)
\(Q_{\mathbf{3a}}\)
\(I^\prime_{\mathbf{3a}}\)
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.185179977455&{}:{}&0.339508342727&{}:{}&0.475311679818&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-1.017825533194&{}:{}&-0.357100710925&{}:{}&2.374926244119&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.022334958066&{}:{}&0.037224930110&{}:{}&0.985110027956&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.043710736551&{}:{}&0.028757730355&{}:{}&1.014953006197&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.001830453876&{}:{}&0.067093486246&{}:{}&0.934736967630&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.015435755462&{}:{}&0.223340349632&{}:{}&0.792095405830&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.023388886457&{}:{}&-0.008205913203&{}:{}&1.031594799660&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.019473481378&{}:{}&0.035702614724&{}:{}&0.944823903898&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.314385271932&{}:{}&-0.121335320020&{}:{}&0.806950048088&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.273462314284&{}:{}&0.726537685716&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.024912518769&{}:{}&0.000000000000&{}:{}&0.975087481231&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.063564219528&{}:{}&0.936435780472&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.611115016909&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&3.053476599582&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.067004874198&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.185179977455&{}:{}&0.339508342727&{}:{}&0.475311679818&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-1.017825533194&{}:{}&-0.357100710925&{}:{}&2.374926244119&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.022334958066&{}:{}&0.037224930110&{}:{}&0.985110027956&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.001907932339&{}:{}&0.027605922934&{}:{}&0.974302009405&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.043710736551&{}:{}&0.028757730355&{}:{}&1.014953006197&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.001830453876&{}:{}&0.067093486246&{}:{}&0.934736967630&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.015435755462&{}:{}&0.223340349632&{}:{}&0.792095405830&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.020353108133&{}:{}&-0.007855173601&{}:{}&0.987502065469&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.023388886457&{}:{}&-0.008205913203&{}:{}&1.031594799660&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.019473481378&{}:{}&0.035702614724&{}:{}&0.944823903898&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.314385271932&{}:{}&-0.121335320020&{}:{}&0.806950048088&. \end{alignedat} \]
3a (033)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.021532679463&{}:{}&0.058867491912&{}:{}&0.962665187551&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.185179977455&{}:{}&0.339508342727&{}:{}&0.475311679818&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-1.017825533194&{}:{}&-0.357100710925&{}:{}&2.374926244119&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.022334958066&{}:{}&0.037224930110&{}:{}&0.985110027956&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.043710736551&{}:{}&0.028757730355&{}:{}&1.014953006197&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.001830453876&{}:{}&0.067093486246&{}:{}&0.934736967630&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.015435755462&{}:{}&0.223340349632&{}:{}&0.792095405830&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.018224037448&{}:{}&0.268478726828&{}:{}&0.713297235724&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.273462314284&{}:{}&0.726537685716&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.024912518769&{}:{}&0.000000000000&{}:{}&0.975087481231&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.063564219528&{}:{}&0.936435780472&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.026021777990&{}:{}&0.127785328297&{}:{}&0.898236449692&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.043710736551&{}:{}&0.028757730355&{}:{}&1.014953006197&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.001830453876&{}:{}&0.067093486246&{}:{}&0.934736967630&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.015435755462&{}:{}&0.223340349632&{}:{}&0.792095405830&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.273462314284&{}:{}&0.726537685716&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.024912518769&{}:{}&0.000000000000&{}:{}&0.975087481231&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.063564219528&{}:{}&0.936435780472&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)