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 > 4c (310)

4c(310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&, \\ P_{\mathbf{4c}}&{}\approx{}&0.196499725902&{}:{}&0.985806327657&{}:{}&-0.182306053559&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.073110737048&{}:{}&0.926889262952&{}:{}&0.000000000000&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.302562013681&{}:{}&1.036450057944&{}:{}&-0.339012071625&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.537708643251&{}:{}&1.204375309918&{}:{}&0.333333333333&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.477469104681&{}:{}&1.069449055997&{}:{}&-0.546918160678&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{4c}}\)
\(P^-_{\mathbf{4c}}\)
\(P^+_{\mathbf{4c}}\)
\(Q_{\mathbf{4c}}\)
\(I^\prime_{\mathbf{4c}}\)
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&4.761972627396&{}:{}&9.082205338335&{}:{}&-12.844177965731&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.160435348784&{}:{}&1.066454497351&{}:{}&-0.226889846135&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.242923624666&{}:{}&0.928849317646&{}:{}&-0.171772942312&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.722389965405&{}:{}&0.947819945164&{}:{}&-0.670209910570&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.240309625328&{}:{}&1.205593280895&{}:{}&-0.445902906223&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.130765894990&{}:{}&0.869234105010&{}:{}&0.000000000000&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.343969330587&{}:{}&0.656030669413&{}:{}&0.000000000000&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.091069351453&{}:{}&1.154566448836&{}:{}&-0.245635800289&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.783227248675&{}:{}&0.216772751325&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.630986313698&{}:{}&0.000000000000&{}:{}&-1.630986313698&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.806562964876&{}:{}&1.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&5.320232710939&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.093980851434&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.292893218813&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&4.761972627396&{}:{}&9.082205338335&{}:{}&-12.844177965731&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.160435348784&{}:{}&1.066454497351&{}:{}&-0.226889846135&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.196499725902&{}:{}&0.985806327657&{}:{}&-0.182306053559&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.242923624666&{}:{}&0.928849317646&{}:{}&-0.171772942312&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.722389965405&{}:{}&0.947819945164&{}:{}&-0.670209910570&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.240309625328&{}:{}&1.205593280895&{}:{}&-0.445902906223&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.073110737048&{}:{}&0.926889262952&{}:{}&0.000000000000&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.130765894990&{}:{}&0.869234105010&{}:{}&0.000000000000&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.343969330587&{}:{}&0.656030669413&{}:{}&0.000000000000&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.091069351453&{}:{}&1.154566448836&{}:{}&-0.245635800289&. \end{alignedat} \]
4c (310)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.302562013681&{}:{}&1.036450057944&{}:{}&-0.339012071625&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&4.761972627396&{}:{}&9.082205338335&{}:{}&-12.844177965731&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.160435348784&{}:{}&1.066454497351&{}:{}&-0.226889846135&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.242923624666&{}:{}&0.928849317646&{}:{}&-0.171772942312&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.722389965405&{}:{}&0.947819945164&{}:{}&-0.670209910570&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.240309625328&{}:{}&1.205593280895&{}:{}&-0.445902906223&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.537708643251&{}:{}&1.204375309918&{}:{}&0.333333333333&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.783227248675&{}:{}&0.216772751325&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.630986313698&{}:{}&0.000000000000&{}:{}&-1.630986313698&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.806562964876&{}:{}&1.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.477469104681&{}:{}&1.069449055997&{}:{}&-0.546918160678&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.242923624666&{}:{}&0.928849317646&{}:{}&-0.171772942312&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.722389965405&{}:{}&0.947819945164&{}:{}&-0.670209910570&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.240309625328&{}:{}&1.205593280895&{}:{}&-0.445902906223&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.783227248675&{}:{}&0.216772751325&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.630986313698&{}:{}&0.000000000000&{}:{}&-1.630986313698&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.806562964876&{}:{}&1.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)