Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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7c(332)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.092106566391&{}:{}&-0.123178333770&{}:{}&1.215284900161&, \\ P^-_{\mathbf{7c}}&{}\approx{}&-0.009343125490&{}:{}&0.013948850609&{}:{}&0.995394274881&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.179551094354&{}:{}&-0.268061409903&{}:{}&1.447612504257&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.252581090427&{}:{}&0.217714045334&{}:{}&0.529704864240&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.327264710139&{}:{}&-0.488590949150&{}:{}&1.815855659289&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{7c}}\)
\(P^-_{\mathbf{7c}}\)
\(P^+_{\mathbf{7c}}\)
\(Q_{\mathbf{7c}}\)
\(I^\prime_{\mathbf{7c}}\)
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&-0.211435789111&{}:{}&-3.028589472778&{}:{}&4.240025261889&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.663549471230&{}:{}&0.115267371694&{}:{}&1.548282099536&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.102779483315&{}:{}&-0.171299138859&{}:{}&1.274078622174&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.177989547185&{}:{}&-0.132865046404&{}:{}&1.310854593588&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.108596164417&{}:{}&-0.324258017455&{}:{}&1.432854181872&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.542616100337&{}:{}&-0.725665386687&{}:{}&2.268281487024&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.086430810133&{}:{}&0.015014181684&{}:{}&1.071416628449&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.010963266475&{}:{}&-0.157036959415&{}:{}&1.168000225890&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.048670226433&{}:{}&0.072662378165&{}:{}&0.976007848268&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.291287847478&{}:{}&0.708712152522&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.322875655532&{}:{}&0.000000000000&{}:{}&0.677124344468&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.537069323539&{}:{}&0.462930676461&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.605717894556&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.221183157077&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.034259827772&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&-0.211435789111&{}:{}&-3.028589472778&{}:{}&4.240025261889&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.663549471230&{}:{}&0.115267371694&{}:{}&1.548282099536&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.102779483315&{}:{}&-0.171299138859&{}:{}&1.274078622174&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.092106566391&{}:{}&-0.123178333770&{}:{}&1.215284900161&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.177989547185&{}:{}&-0.132865046404&{}:{}&1.310854593588&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.108596164417&{}:{}&-0.324258017455&{}:{}&1.432854181872&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.542616100337&{}:{}&-0.725665386687&{}:{}&2.268281487024&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&-0.009343125490&{}:{}&0.013948850609&{}:{}&0.995394274881&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.086430810133&{}:{}&0.015014181684&{}:{}&1.071416628449&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.010963266475&{}:{}&-0.157036959415&{}:{}&1.168000225890&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.048670226433&{}:{}&0.072662378165&{}:{}&0.976007848268&. \end{alignedat} \]
7c (332)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.179551094354&{}:{}&-0.268061409903&{}:{}&1.447612504257&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&-0.211435789111&{}:{}&-3.028589472778&{}:{}&4.240025261889&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.663549471230&{}:{}&0.115267371694&{}:{}&1.548282099536&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.102779483315&{}:{}&-0.171299138859&{}:{}&1.274078622174&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.177989547185&{}:{}&-0.132865046404&{}:{}&1.310854593588&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.108596164417&{}:{}&-0.324258017455&{}:{}&1.432854181872&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.542616100337&{}:{}&-0.725665386687&{}:{}&2.268281487024&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.252581090427&{}:{}&0.217714045334&{}:{}&0.529704864240&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.291287847478&{}:{}&0.708712152522&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.322875655532&{}:{}&0.000000000000&{}:{}&0.677124344468&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.537069323539&{}:{}&0.462930676461&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.327264710139&{}:{}&-0.488590949150&{}:{}&1.815855659289&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.177989547185&{}:{}&-0.132865046404&{}:{}&1.310854593588&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.108596164417&{}:{}&-0.324258017455&{}:{}&1.432854181872&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.542616100337&{}:{}&-0.725665386687&{}:{}&2.268281487024&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.291287847478&{}:{}&0.708712152522&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.322875655532&{}:{}&0.000000000000&{}:{}&0.677124344468&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.537069323539&{}:{}&0.462930676461&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)