Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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6b(321)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.002779412860&{}:{}&1.004004467550&{}:{}&-0.001225054690&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.008037794643&{}:{}&0.968041507137&{}:{}&0.023920698220&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.013999314970&{}:{}&1.041306229146&{}:{}&-0.027306914176&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.242210876421&{}:{}&1.188609789827&{}:{}&0.053601086594&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.044399942077&{}:{}&1.089426837486&{}:{}&-0.045026895409&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{6b}}\)
\(P^-_{\mathbf{6b}}\)
\(P^+_{\mathbf{6b}}\)
\(Q_{\mathbf{6b}}\)
\(I^\prime_{\mathbf{6b}}\)
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&9.350893551327&{}:{}&20.877233878318&{}:{}&-29.228127429646&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.011551064408&{}:{}&1.038503548026&{}:{}&-0.026952483619&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.346228427385&{}:{}&0.577047378975&{}:{}&0.769181048410&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.022356266674&{}:{}&1.023605237608&{}:{}&-0.001248970933&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.085408745089&{}:{}&1.123053524400&{}:{}&-0.037644779311&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.002925515071&{}:{}&1.056780820070&{}:{}&-0.053855304999&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.010973691035&{}:{}&0.986594539852&{}:{}&0.024379151183&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.161065002739&{}:{}&0.359601113342&{}:{}&0.479333883919&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.008452334902&{}:{}&1.017967163966&{}:{}&-0.026419498867&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.956850251748&{}:{}&0.043149748252&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.284190373377&{}:{}&0.000000000000&{}:{}&-0.284190373377&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.255928946018&{}:{}&1.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-20.877233878318&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.019251774013&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.577047378975&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-1.000000000000&{}:{}&1.400000000000&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&9.350893551327&{}:{}&20.877233878318&{}:{}&-29.228127429646&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.011551064408&{}:{}&1.038503548026&{}:{}&-0.026952483619&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.346228427385&{}:{}&0.577047378975&{}:{}&0.769181048410&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.002779412860&{}:{}&1.004004467550&{}:{}&-0.001225054690&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.022356266674&{}:{}&1.023605237608&{}:{}&-0.001248970933&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.085408745089&{}:{}&1.123053524400&{}:{}&-0.037644779311&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.002925515071&{}:{}&1.056780820070&{}:{}&-0.053855304999&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.008037794643&{}:{}&0.968041507137&{}:{}&0.023920698220&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.010973691035&{}:{}&0.986594539852&{}:{}&0.024379151183&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.161065002739&{}:{}&0.359601113342&{}:{}&0.479333883919&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.008452334902&{}:{}&1.017967163966&{}:{}&-0.026419498867&. \end{alignedat} \]
6b (321)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.013999314970&{}:{}&1.041306229146&{}:{}&-0.027306914176&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&9.350893551327&{}:{}&20.877233878318&{}:{}&-29.228127429646&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.011551064408&{}:{}&1.038503548026&{}:{}&-0.026952483619&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.346228427385&{}:{}&0.577047378975&{}:{}&0.769181048410&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.022356266674&{}:{}&1.023605237608&{}:{}&-0.001248970933&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.085408745089&{}:{}&1.123053524400&{}:{}&-0.037644779311&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.002925515071&{}:{}&1.056780820070&{}:{}&-0.053855304999&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.242210876421&{}:{}&1.188609789827&{}:{}&0.053601086594&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.956850251748&{}:{}&0.043149748252&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.284190373377&{}:{}&0.000000000000&{}:{}&-0.284190373377&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.255928946018&{}:{}&1.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.044399942077&{}:{}&1.089426837486&{}:{}&-0.045026895409&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.022356266674&{}:{}&1.023605237608&{}:{}&-0.001248970933&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.085408745089&{}:{}&1.123053524400&{}:{}&-0.037644779311&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.002925515071&{}:{}&1.056780820070&{}:{}&-0.053855304999&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.956850251748&{}:{}&0.043149748252&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.284190373377&{}:{}&0.000000000000&{}:{}&-0.284190373377&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.255928946018&{}:{}&1.255928946018&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)