Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).

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2c (130)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&, \\ P_{\mathbf{2c}}&{}\approx{}&0.938872591365&{}:{}&0.602436420515&{}:{}&-0.541309011880&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.898939327946&{}:{}&0.294708668716&{}:{}&-0.193647996661&, \\ P^+_{\mathbf{2c}}&{}\approx{}&0.970648871349&{}:{}&0.847306044559&{}:{}&-0.817954915908&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.110089629930&{}:{}&-0.487047089356&{}:{}&0.376957459426&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&0.983662115547&{}:{}&1.220685668015&{}:{}&-1.204347783562&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{2c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2c}}\) Radical center of the Malfatti circles
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.037016188229&{}:{}&0.444194258749&{}:{}&-0.481210446978&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&4.221126147106&{}:{}&7.753801835369&{}:{}&-10.974927982475&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.455995886339&{}:{}&1.094390127214&{}:{}&-0.550386013553&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.823077159430&{}:{}&1.743649291855&{}:{}&-1.566726451285&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.869273759103&{}:{}&0.631907896343&{}:{}&-0.501181655446&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.170763435259&{}:{}&0.751231359500&{}:{}&-0.921994794759&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.613532372747&{}:{}&1.126999827078&{}:{}&-0.740532199825&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.824456361540&{}:{}&0.353146639890&{}:{}&-0.177603001430&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.157666333777&{}:{}&0.379529845272&{}:{}&-0.537196179049&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.424095981292&{}:{}&-3.424095981292&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.746506037284&{}:{}&0.000000000000&{}:{}&0.253493962716&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.781723650696&{}:{}&-0.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.148064752916&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&3.376900917685&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.364796709071&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.037016188229&{}:{}&0.444194258749&{}:{}&-0.481210446978&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&4.221126147106&{}:{}&7.753801835369&{}:{}&-10.974927982475&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.455995886339&{}:{}&1.094390127214&{}:{}&-0.550386013553&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&0.938872591365&{}:{}&0.602436420515&{}:{}&-0.541309011880&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.823077159430&{}:{}&1.743649291855&{}:{}&-1.566726451285&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.869273759103&{}:{}&0.631907896343&{}:{}&-0.501181655446&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.170763435259&{}:{}&0.751231359500&{}:{}&-0.921994794759&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.898939327946&{}:{}&0.294708668716&{}:{}&-0.193647996661&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.613532372747&{}:{}&1.126999827078&{}:{}&-0.740532199825&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.824456361540&{}:{}&0.353146639890&{}:{}&-0.177603001430&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.157666333777&{}:{}&0.379529845272&{}:{}&-0.537196179049&. \end{alignedat} \]
2c (130)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&0.970648871349&{}:{}&0.847306044559&{}:{}&-0.817954915908&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.037016188229&{}:{}&0.444194258749&{}:{}&-0.481210446978&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&4.221126147106&{}:{}&7.753801835369&{}:{}&-10.974927982475&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.455995886339&{}:{}&1.094390127214&{}:{}&-0.550386013553&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.823077159430&{}:{}&1.743649291855&{}:{}&-1.566726451285&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.869273759103&{}:{}&0.631907896343&{}:{}&-0.501181655446&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.170763435259&{}:{}&0.751231359500&{}:{}&-0.921994794759&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.110089629930&{}:{}&-0.487047089356&{}:{}&0.376957459426&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.424095981292&{}:{}&-3.424095981292&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.746506037284&{}:{}&0.000000000000&{}:{}&0.253493962716&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.781723650696&{}:{}&-0.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&0.983662115547&{}:{}&1.220685668015&{}:{}&-1.204347783562&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.823077159430&{}:{}&1.743649291855&{}:{}&-1.566726451285&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.869273759103&{}:{}&0.631907896343&{}:{}&-0.501181655446&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.170763435259&{}:{}&0.751231359500&{}:{}&-0.921994794759&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.424095981292&{}:{}&-3.424095981292&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.746506037284&{}:{}&0.000000000000&{}:{}&0.253493962716&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.781723650696&{}:{}&-0.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)