Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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0b(101)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&, \\ P_{\mathbf{b}}&{}\approx{}&0.755037788899&{}:{}&-0.470293118044&{}:{}&0.715255329145&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.795294173629&{}:{}&-0.332751914798&{}:{}&0.537457741169&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.728541307969&{}:{}&-0.560821810409&{}:{}&0.832280502439&, \\ Q_{\mathbf{b}}&{}\approx{}&0.892334222867&{}:{}&-0.181836845053&{}:{}&0.289502622186&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.681265979294&{}:{}&-0.694130366975&{}:{}&1.012864387681&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{b}}\)
\(P^-_{\mathbf{b}}\)
\(P^+_{\mathbf{b}}\)
\(Q_{\mathbf{b}}\)
\(I^\prime_{\mathbf{b}}\)
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.884489355921&{}:{}&-0.288776610198&{}:{}&0.404287254278&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.835089355133&{}:{}&-1.783631183776&{}:{}&1.948541828643&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.359305255565&{}:{}&-0.598842092609&{}:{}&1.239536837044&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.502433603565&{}:{}&-0.955257755724&{}:{}&1.452824152159&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.713626705381&{}:{}&-0.389652874598&{}:{}&0.676026169217&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.876000940496&{}:{}&-0.545637873723&{}:{}&0.669636933227&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.432159811138&{}:{}&-0.923031422664&{}:{}&1.490871611526&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.741120760348&{}:{}&-0.241968249237&{}:{}&0.500847488889&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.962756304520&{}:{}&-0.402818245670&{}:{}&0.440061941150&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.688901059317&{}:{}&2.688901059317&,\\B^*_{\mathbf{b}}&{}\approx{}&0.755040111164&{}:{}&0.000000000000&{}:{}&0.244959888836&,\\C^*_{\mathbf{b}}&{}\approx{}&1.255928946018&{}:{}&-0.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.288776610198&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.391815591888&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.598842092609&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.884489355921&{}:{}&-0.288776610198&{}:{}&0.404287254278&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.835089355133&{}:{}&-1.783631183776&{}:{}&1.948541828643&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.359305255565&{}:{}&-0.598842092609&{}:{}&1.239536837044&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.755037788899&{}:{}&-0.470293118044&{}:{}&0.715255329145&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.502433603565&{}:{}&-0.955257755724&{}:{}&1.452824152159&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.713626705381&{}:{}&-0.389652874598&{}:{}&0.676026169217&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.876000940496&{}:{}&-0.545637873723&{}:{}&0.669636933227&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.795294173629&{}:{}&-0.332751914798&{}:{}&0.537457741169&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.432159811138&{}:{}&-0.923031422664&{}:{}&1.490871611526&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.741120760348&{}:{}&-0.241968249237&{}:{}&0.500847488889&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.962756304520&{}:{}&-0.402818245670&{}:{}&0.440061941150&. \end{alignedat} \]
0b (101)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.728541307969&{}:{}&-0.560821810409&{}:{}&0.832280502439&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.884489355921&{}:{}&-0.288776610198&{}:{}&0.404287254278&,\\B^\prime_{\mathbf{b}}&{}\approx{}&0.835089355133&{}:{}&-1.783631183776&{}:{}&1.948541828643&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.359305255565&{}:{}&-0.598842092609&{}:{}&1.239536837044&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.502433603565&{}:{}&-0.955257755724&{}:{}&1.452824152159&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.713626705381&{}:{}&-0.389652874598&{}:{}&0.676026169217&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.876000940496&{}:{}&-0.545637873723&{}:{}&0.669636933227&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.892334222867&{}:{}&-0.181836845053&{}:{}&0.289502622186&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.688901059317&{}:{}&2.688901059317&,\\B^*_{\mathbf{b}}&{}\approx{}&0.755040111164&{}:{}&0.000000000000&{}:{}&0.244959888836&,\\C^*_{\mathbf{b}}&{}\approx{}&1.255928946018&{}:{}&-0.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.681265979294&{}:{}&-0.694130366975&{}:{}&1.012864387681&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.502433603565&{}:{}&-0.955257755724&{}:{}&1.452824152159&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.713626705381&{}:{}&-0.389652874598&{}:{}&0.676026169217&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.876000940496&{}:{}&-0.545637873723&{}:{}&0.669636933227&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.688901059317&{}:{}&2.688901059317&,\\B^*_{\mathbf{b}}&{}\approx{}&0.755040111164&{}:{}&0.000000000000&{}:{}&0.244959888836&,\\C^*_{\mathbf{b}}&{}\approx{}&1.255928946018&{}:{}&-0.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)