Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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1c(112)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.368149323909&{}:{}&-0.891918732013&{}:{}&2.260068055923&, \\ P^-_{\mathbf{1c}}&{}\approx{}&0.065956458802&{}:{}&-0.132535537606&{}:{}&1.066579078804&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-0.953017124550&{}:{}&-1.915030601159&{}:{}&3.868047725708&, \\ Q_{\mathbf{1c}}&{}\approx{}&1.370520539954&{}:{}&1.590010290344&{}:{}&-1.960530830298&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-1.289807158786&{}:{}&-2.591789921755&{}:{}&4.881597080540&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{1c}}\)
\(P^-_{\mathbf{1c}}\)
\(P^+_{\mathbf{1c}}\)
\(Q_{\mathbf{1c}}\)
\(I^\prime_{\mathbf{1c}}\)
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&0.203740634524&{}:{}&-1.990648413689&{}:{}&2.786907779164&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.009529824259&{}:{}&-0.346039765679&{}:{}&2.355569589938&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-10.893372939391&{}:{}&-18.155621565652&{}:{}&30.048994505044&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.847815627408&{}:{}&-1.856537191531&{}:{}&4.704352818939&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.892762666339&{}:{}&-3.587905781692&{}:{}&5.480668448031&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.453777646476&{}:{}&-1.099371251760&{}:{}&2.553148898235&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.706368134193&{}:{}&-0.242124063862&{}:{}&1.948492198055&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.135126833464&{}:{}&-1.320257087198&{}:{}&2.185130253734&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.078931184156&{}:{}&-0.158607467957&{}:{}&1.079676283801&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-4.291287847478&{}:{}&5.291287847478&,\\B^*_{\mathbf{1c}}&{}\approx{}&-2.322875655532&{}:{}&0.000000000000&{}:{}&3.322875655532&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.462930676461&{}:{}&0.537069323539&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.398129682738&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.336509941420&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-3.631124313130&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.203740634524&{}:{}&-1.990648413689&{}:{}&2.786907779164&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.009529824259&{}:{}&-0.346039765679&{}:{}&2.355569589938&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-10.893372939391&{}:{}&-18.155621565652&{}:{}&30.048994505044&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.368149323909&{}:{}&-0.891918732013&{}:{}&2.260068055923&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.847815627408&{}:{}&-1.856537191531&{}:{}&4.704352818939&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.892762666339&{}:{}&-3.587905781692&{}:{}&5.480668448031&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.453777646476&{}:{}&-1.099371251760&{}:{}&2.553148898235&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&0.065956458802&{}:{}&-0.132535537606&{}:{}&1.066579078804&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.706368134193&{}:{}&-0.242124063862&{}:{}&1.948492198055&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.135126833464&{}:{}&-1.320257087198&{}:{}&2.185130253734&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.078931184156&{}:{}&-0.158607467957&{}:{}&1.079676283801&. \end{alignedat} \]
1c (112)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-0.953017124550&{}:{}&-1.915030601159&{}:{}&3.868047725708&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.203740634524&{}:{}&-1.990648413689&{}:{}&2.786907779164&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.009529824259&{}:{}&-0.346039765679&{}:{}&2.355569589938&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-10.893372939391&{}:{}&-18.155621565652&{}:{}&30.048994505044&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.847815627408&{}:{}&-1.856537191531&{}:{}&4.704352818939&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.892762666339&{}:{}&-3.587905781692&{}:{}&5.480668448031&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.453777646476&{}:{}&-1.099371251760&{}:{}&2.553148898235&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&1.370520539954&{}:{}&1.590010290344&{}:{}&-1.960530830298&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-4.291287847478&{}:{}&5.291287847478&,\\B^*_{\mathbf{1c}}&{}\approx{}&-2.322875655532&{}:{}&0.000000000000&{}:{}&3.322875655532&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.462930676461&{}:{}&0.537069323539&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-1.289807158786&{}:{}&-2.591789921755&{}:{}&4.881597080540&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.847815627408&{}:{}&-1.856537191531&{}:{}&4.704352818939&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.892762666339&{}:{}&-3.587905781692&{}:{}&5.480668448031&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.453777646476&{}:{}&-1.099371251760&{}:{}&2.553148898235&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-4.291287847478&{}:{}&5.291287847478&,\\B^*_{\mathbf{1c}}&{}\approx{}&-2.322875655532&{}:{}&0.000000000000&{}:{}&3.322875655532&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.462930676461&{}:{}&0.537069323539&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)