Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 90° Top Angle

\(A=45\degree\), \(B=45\degree\), \(C=90\degree\).


[Top] > Isosceles Triangle with 90° Top Angle > 2c (130)

2c(130)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&, \\ P_{\mathbf{2c}}&{}\approx{}&0.985806327657&{}:{}&0.196499725902&{}:{}&-0.182306053559&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.926889262952&{}:{}&0.073110737048&{}:{}&0.000000000000&, \\ P^+_{\mathbf{2c}}&{}\approx{}&1.036450057944&{}:{}&0.302562013681&{}:{}&-0.339012071625&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.204375309918&{}:{}&-0.537708643251&{}:{}&0.333333333333&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&1.069449055997&{}:{}&0.477469104681&{}:{}&-0.546918160678&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{2c}}\)
\(P^-_{\mathbf{2c}}\)
\(P^+_{\mathbf{2c}}\)
\(Q_{\mathbf{2c}}\)
\(I^\prime_{\mathbf{2c}}\)
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.066454497351&{}:{}&0.160435348784&{}:{}&-0.226889846135&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&9.082205338335&{}:{}&4.761972627396&{}:{}&-12.844177965731&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.947819945164&{}:{}&0.722389965405&{}:{}&-0.670209910570&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.928849317646&{}:{}&0.242923624666&{}:{}&-0.171772942312&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.205593280895&{}:{}&0.240309625328&{}:{}&-0.445902906223&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.656030669413&{}:{}&0.343969330587&{}:{}&0.000000000000&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.869234105010&{}:{}&0.130765894990&{}:{}&0.000000000000&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.154566448836&{}:{}&0.091069351453&{}:{}&-0.245635800289&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&2.630986313698&{}:{}&-1.630986313698&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.783227248675&{}:{}&0.000000000000&{}:{}&0.216772751325&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.806562964876&{}:{}&-0.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.093980851434&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&5.320232710939&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.292893218813&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.066454497351&{}:{}&0.160435348784&{}:{}&-0.226889846135&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&9.082205338335&{}:{}&4.761972627396&{}:{}&-12.844177965731&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&0.985806327657&{}:{}&0.196499725902&{}:{}&-0.182306053559&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.947819945164&{}:{}&0.722389965405&{}:{}&-0.670209910570&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.928849317646&{}:{}&0.242923624666&{}:{}&-0.171772942312&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.205593280895&{}:{}&0.240309625328&{}:{}&-0.445902906223&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.926889262952&{}:{}&0.073110737048&{}:{}&0.000000000000&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.656030669413&{}:{}&0.343969330587&{}:{}&0.000000000000&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.869234105010&{}:{}&0.130765894990&{}:{}&0.000000000000&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.154566448836&{}:{}&0.091069351453&{}:{}&-0.245635800289&. \end{alignedat} \]
2c (130)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&1.036450057944&{}:{}&0.302562013681&{}:{}&-0.339012071625&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.066454497351&{}:{}&0.160435348784&{}:{}&-0.226889846135&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&9.082205338335&{}:{}&4.761972627396&{}:{}&-12.844177965731&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.947819945164&{}:{}&0.722389965405&{}:{}&-0.670209910570&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.928849317646&{}:{}&0.242923624666&{}:{}&-0.171772942312&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.205593280895&{}:{}&0.240309625328&{}:{}&-0.445902906223&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.204375309918&{}:{}&-0.537708643251&{}:{}&0.333333333333&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&2.630986313698&{}:{}&-1.630986313698&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.783227248675&{}:{}&0.000000000000&{}:{}&0.216772751325&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.806562964876&{}:{}&-0.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&1.069449055997&{}:{}&0.477469104681&{}:{}&-0.546918160678&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.947819945164&{}:{}&0.722389965405&{}:{}&-0.670209910570&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.928849317646&{}:{}&0.242923624666&{}:{}&-0.171772942312&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.205593280895&{}:{}&0.240309625328&{}:{}&-0.445902906223&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&2.630986313698&{}:{}&-1.630986313698&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.783227248675&{}:{}&0.000000000000&{}:{}&0.216772751325&,\\C^*_{\mathbf{2c}}&{}\approx{}&1.806562964876&{}:{}&-0.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)