Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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0a(011)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&, \\ P_{\mathbf{a}}&{}\approx{}&-0.186969427939&{}:{}&0.694804074166&{}:{}&0.492165353773&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.144617378745&{}:{}&0.735097205827&{}:{}&0.409520172919&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.213796211484&{}:{}&0.669281457387&{}:{}&0.544514754096&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.107802540284&{}:{}&0.804860293848&{}:{}&0.302942246436&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.252078775602&{}:{}&0.627345134507&{}:{}&0.624733641096&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-0.599580303007&{}:{}&0.666491792920&{}:{}&0.933088510088&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.111674790330&{}:{}&0.851100279560&{}:{}&0.260574510770&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.203565106639&{}:{}&0.339275177731&{}:{}&0.864289928907&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.144227242941&{}:{}&0.669784521365&{}:{}&0.474442721576&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.337152836255&{}:{}&0.449655095261&{}:{}&0.887497740994&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.218183694473&{}:{}&0.810800575833&{}:{}&0.407383118640&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.092021668855&{}:{}&0.701319141561&{}:{}&0.390702527294&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.339749318725&{}:{}&0.377664395319&{}:{}&0.962084923406&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.177327445463&{}:{}&0.901364073993&{}:{}&0.275963371470&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.726537685716&{}:{}&0.273462314284&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.552437750421&{}:{}&0.000000000000&{}:{}&1.552437750421&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.154653670708&{}:{}&1.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.199685227256&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.335024370989&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.610695319916&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.333333333333&{}:{}&0.555555555556&{}:{}&0.777777777778&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.599580303007&{}:{}&0.666491792920&{}:{}&0.933088510088&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.111674790330&{}:{}&0.851100279560&{}:{}&0.260574510770&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.203565106639&{}:{}&0.339275177731&{}:{}&0.864289928907&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.186969427939&{}:{}&0.694804074166&{}:{}&0.492165353773&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.144227242941&{}:{}&0.669784521365&{}:{}&0.474442721576&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.337152836255&{}:{}&0.449655095261&{}:{}&0.887497740994&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.218183694473&{}:{}&0.810800575833&{}:{}&0.407383118640&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.144617378745&{}:{}&0.735097205827&{}:{}&0.409520172919&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.092021668855&{}:{}&0.701319141561&{}:{}&0.390702527294&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.339749318725&{}:{}&0.377664395319&{}:{}&0.962084923406&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.177327445463&{}:{}&0.901364073993&{}:{}&0.275963371470&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.213796211484&{}:{}&0.669281457387&{}:{}&0.544514754096&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.599580303007&{}:{}&0.666491792920&{}:{}&0.933088510088&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.111674790330&{}:{}&0.851100279560&{}:{}&0.260574510770&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.203565106639&{}:{}&0.339275177731&{}:{}&0.864289928907&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.144227242941&{}:{}&0.669784521365&{}:{}&0.474442721576&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.337152836255&{}:{}&0.449655095261&{}:{}&0.887497740994&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.218183694473&{}:{}&0.810800575833&{}:{}&0.407383118640&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.107802540284&{}:{}&0.804860293848&{}:{}&0.302942246436&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.726537685716&{}:{}&0.273462314284&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.552437750421&{}:{}&0.000000000000&{}:{}&1.552437750421&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.154653670708&{}:{}&1.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.252078775602&{}:{}&0.627345134507&{}:{}&0.624733641096&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.144227242941&{}:{}&0.669784521365&{}:{}&0.474442721576&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.337152836255&{}:{}&0.449655095261&{}:{}&0.887497740994&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.218183694473&{}:{}&0.810800575833&{}:{}&0.407383118640&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.726537685716&{}:{}&0.273462314284&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.552437750421&{}:{}&0.000000000000&{}:{}&1.552437750421&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.154653670708&{}:{}&1.154653670708&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)