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 > 6c (330)

6c(330)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&, \\ P_{\mathbf{6c}}&{}\approx{}&0.550943212689&{}:{}&0.550943212689&{}:{}&-0.101886425379&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.430245455683&{}:{}&0.430245455683&{}:{}&0.139509088634&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.650793266398&{}:{}&0.650793266398&{}:{}&-0.301586532797&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.439218977423&{}:{}&0.439218977423&{}:{}&0.121562045155&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.671772942312&{}:{}&0.671772942312&{}:{}&-0.343545884624&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{6c}}\)
\(P^-_{\mathbf{6c}}\)
\(P^+_{\mathbf{6c}}\)
\(Q_{\mathbf{6c}}\)
\(I^\prime_{\mathbf{6c}}\)
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.207106781187&{}:{}&0.500000000000&{}:{}&-0.707106781187&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&1.207106781187&{}:{}&-0.707106781187&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.359347716164&{}:{}&0.359347716164&{}:{}&0.281304567672&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.418163345762&{}:{}&0.713849483194&{}:{}&-0.132012828956&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.713849483194&{}:{}&0.418163345762&{}:{}&-0.132012828956&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.853553390593&{}:{}&0.853553390593&{}:{}&-0.707106781187&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.238263524168&{}:{}&0.575219031465&{}:{}&0.186517444367&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.575219031465&{}:{}&0.238263524168&{}:{}&0.186517444367&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.707106781187&{}:{}&0.707106781187&{}:{}&-0.414213562373&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.783227248675&{}:{}&0.216772751325&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.783227248675&{}:{}&0.000000000000&{}:{}&0.216772751325&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.292893218813&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.292893218813&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.210501018521&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.207106781187&{}:{}&0.500000000000&{}:{}&-0.707106781187&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&1.207106781187&{}:{}&-0.707106781187&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.359347716164&{}:{}&0.359347716164&{}:{}&0.281304567672&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.550943212689&{}:{}&0.550943212689&{}:{}&-0.101886425379&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.418163345762&{}:{}&0.713849483194&{}:{}&-0.132012828956&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.713849483194&{}:{}&0.418163345762&{}:{}&-0.132012828956&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.853553390593&{}:{}&0.853553390593&{}:{}&-0.707106781187&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.430245455683&{}:{}&0.430245455683&{}:{}&0.139509088634&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.238263524168&{}:{}&0.575219031465&{}:{}&0.186517444367&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.575219031465&{}:{}&0.238263524168&{}:{}&0.186517444367&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.707106781187&{}:{}&0.707106781187&{}:{}&-0.414213562373&. \end{alignedat} \]
6c (330)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.650793266398&{}:{}&0.650793266398&{}:{}&-0.301586532797&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.207106781187&{}:{}&0.500000000000&{}:{}&-0.707106781187&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&1.207106781187&{}:{}&-0.707106781187&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.359347716164&{}:{}&0.359347716164&{}:{}&0.281304567672&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.418163345762&{}:{}&0.713849483194&{}:{}&-0.132012828956&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.713849483194&{}:{}&0.418163345762&{}:{}&-0.132012828956&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.853553390593&{}:{}&0.853553390593&{}:{}&-0.707106781187&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.439218977423&{}:{}&0.439218977423&{}:{}&0.121562045155&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.783227248675&{}:{}&0.216772751325&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.783227248675&{}:{}&0.000000000000&{}:{}&0.216772751325&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.671772942312&{}:{}&0.671772942312&{}:{}&-0.343545884624&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.418163345762&{}:{}&0.713849483194&{}:{}&-0.132012828956&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.713849483194&{}:{}&0.418163345762&{}:{}&-0.132012828956&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.853553390593&{}:{}&0.853553390593&{}:{}&-0.707106781187&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.783227248675&{}:{}&0.216772751325&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.783227248675&{}:{}&0.000000000000&{}:{}&0.216772751325&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)