Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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3c(132)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.039928131880&{}:{}&-0.691404056550&{}:{}&1.731332188429&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.124870308247&{}:{}&-0.219798224130&{}:{}&1.094927915883&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.261287317733&{}:{}&-1.324870508220&{}:{}&2.586157825953&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.600368860211&{}:{}&-1.368375379749&{}:{}&1.768006519538&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.315333151301&{}:{}&-2.032833073008&{}:{}&3.348166224310&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{3c}}\)
\(P^-_{\mathbf{3c}}\)
\(P^+_{\mathbf{3c}}\)
\(Q_{\mathbf{3c}}\)
\(I^\prime_{\mathbf{3c}}\)
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.796317145104&{}:{}&-2.444194258749&{}:{}&2.647877113645&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-4.471126147106&{}:{}&-6.153801835369&{}:{}&11.624927982475&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.289329219672&{}:{}&-0.694390127214&{}:{}&1.983719346886&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.543488947812&{}:{}&-1.026200260424&{}:{}&2.569689208236&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.049137456558&{}:{}&-1.081522211006&{}:{}&2.130659667565&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.180893882927&{}:{}&-3.132397098807&{}:{}&4.313290981733&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.223217891182&{}:{}&-0.307224315138&{}:{}&1.530442206320&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.149272711842&{}:{}&-0.458173615268&{}:{}&1.308900903426&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.389872379765&{}:{}&-0.686258069775&{}:{}&1.296385690010&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.424095981292&{}:{}&4.424095981292&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.253493962716&{}:{}&0.000000000000&{}:{}&0.746506037284&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.781723650696&{}:{}&1.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.814731419583&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-3.576900917685&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.231463375738&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.796317145104&{}:{}&-2.444194258749&{}:{}&2.647877113645&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-4.471126147106&{}:{}&-6.153801835369&{}:{}&11.624927982475&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.289329219672&{}:{}&-0.694390127214&{}:{}&1.983719346886&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.039928131880&{}:{}&-0.691404056550&{}:{}&1.731332188429&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.543488947812&{}:{}&-1.026200260424&{}:{}&2.569689208236&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.049137456558&{}:{}&-1.081522211006&{}:{}&2.130659667565&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.180893882927&{}:{}&-3.132397098807&{}:{}&4.313290981733&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.124870308247&{}:{}&-0.219798224130&{}:{}&1.094927915883&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.223217891182&{}:{}&-0.307224315138&{}:{}&1.530442206320&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.149272711842&{}:{}&-0.458173615268&{}:{}&1.308900903426&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.389872379765&{}:{}&-0.686258069775&{}:{}&1.296385690010&. \end{alignedat} \]
3c (132)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.261287317733&{}:{}&-1.324870508220&{}:{}&2.586157825953&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.796317145104&{}:{}&-2.444194258749&{}:{}&2.647877113645&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-4.471126147106&{}:{}&-6.153801835369&{}:{}&11.624927982475&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.289329219672&{}:{}&-0.694390127214&{}:{}&1.983719346886&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.543488947812&{}:{}&-1.026200260424&{}:{}&2.569689208236&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.049137456558&{}:{}&-1.081522211006&{}:{}&2.130659667565&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.180893882927&{}:{}&-3.132397098807&{}:{}&4.313290981733&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.600368860211&{}:{}&-1.368375379749&{}:{}&1.768006519538&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.424095981292&{}:{}&4.424095981292&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.253493962716&{}:{}&0.000000000000&{}:{}&0.746506037284&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.781723650696&{}:{}&1.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.315333151301&{}:{}&-2.032833073008&{}:{}&3.348166224310&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.543488947812&{}:{}&-1.026200260424&{}:{}&2.569689208236&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.049137456558&{}:{}&-1.081522211006&{}:{}&2.130659667565&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.180893882927&{}:{}&-3.132397098807&{}:{}&4.313290981733&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-3.424095981292&{}:{}&4.424095981292&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.253493962716&{}:{}&0.000000000000&{}:{}&0.746506037284&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.781723650696&{}:{}&1.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)