Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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2a(031)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.013299106188&{}:{}&0.032149812806&{}:{}&0.981149293381&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.018395315697&{}:{}&-0.009170524006&{}:{}&0.990775208309&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.041341264414&{}:{}&0.068708656728&{}:{}&0.972632607686&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.082859945762&{}:{}&-0.315571858341&{}:{}&1.232711912580&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.084146438277&{}:{}&0.151071695632&{}:{}&0.933074742645&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{2a}}\)
\(P^-_{\mathbf{2a}}\)
\(P^+_{\mathbf{2a}}\)
\(Q_{\mathbf{2a}}\)
\(I^\prime_{\mathbf{2a}}\)
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.158475478364&{}:{}&0.374010898505&{}:{}&0.467513623131&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-2.597262605646&{}:{}&-0.731508403764&{}:{}&4.328771009409&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.032507527261&{}:{}&0.043343369682&{}:{}&0.989164157580&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.064211380228&{}:{}&0.033765150341&{}:{}&1.030446229887&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.012673425524&{}:{}&0.077684086996&{}:{}&0.934989338528&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.110137665926&{}:{}&0.266251377534&{}:{}&0.843886288393&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.034308699033&{}:{}&-0.009662904941&{}:{}&1.043971603973&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.017476331757&{}:{}&0.041245110035&{}:{}&0.941278558208&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.289725392033&{}:{}&-0.144435339219&{}:{}&0.854709947186&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.344082517041&{}:{}&1.344082517041&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.062983975551&{}:{}&0.000000000000&{}:{}&0.937016024449&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.356062329784&{}:{}&1.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.561016347758&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&5.194525211291&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.065015054523&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.158475478364&{}:{}&0.374010898505&{}:{}&0.467513623131&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-2.597262605646&{}:{}&-0.731508403764&{}:{}&4.328771009409&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.032507527261&{}:{}&0.043343369682&{}:{}&0.989164157580&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.013299106188&{}:{}&0.032149812806&{}:{}&0.981149293381&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.064211380228&{}:{}&0.033765150341&{}:{}&1.030446229887&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.012673425524&{}:{}&0.077684086996&{}:{}&0.934989338528&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.110137665926&{}:{}&0.266251377534&{}:{}&0.843886288393&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.018395315697&{}:{}&-0.009170524006&{}:{}&0.990775208309&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.034308699033&{}:{}&-0.009662904941&{}:{}&1.043971603973&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.017476331757&{}:{}&0.041245110035&{}:{}&0.941278558208&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.289725392033&{}:{}&-0.144435339219&{}:{}&0.854709947186&. \end{alignedat} \]
2a (031)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.041341264414&{}:{}&0.068708656728&{}:{}&0.972632607686&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.158475478364&{}:{}&0.374010898505&{}:{}&0.467513623131&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-2.597262605646&{}:{}&-0.731508403764&{}:{}&4.328771009409&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.032507527261&{}:{}&0.043343369682&{}:{}&0.989164157580&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.064211380228&{}:{}&0.033765150341&{}:{}&1.030446229887&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.012673425524&{}:{}&0.077684086996&{}:{}&0.934989338528&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.110137665926&{}:{}&0.266251377534&{}:{}&0.843886288393&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.082859945762&{}:{}&-0.315571858341&{}:{}&1.232711912580&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.344082517041&{}:{}&1.344082517041&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.062983975551&{}:{}&0.000000000000&{}:{}&0.937016024449&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.356062329784&{}:{}&1.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.084146438277&{}:{}&0.151071695632&{}:{}&0.933074742645&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.064211380228&{}:{}&0.033765150341&{}:{}&1.030446229887&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.012673425524&{}:{}&0.077684086996&{}:{}&0.934989338528&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.110137665926&{}:{}&0.266251377534&{}:{}&0.843886288393&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.344082517041&{}:{}&1.344082517041&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.062983975551&{}:{}&0.000000000000&{}:{}&0.937016024449&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.356062329784&{}:{}&1.356062329784&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)