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 > 7c (332)

7c(332)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.033879030723&{}:{}&-0.033879030723&{}:{}&1.067758061446&, \\ P^-_{\mathbf{7c}}&{}\approx{}&0.023171078678&{}:{}&0.023171078678&{}:{}&0.953657842645&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.094930300373&{}:{}&-0.094930300373&{}:{}&1.189860600745&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.178153850905&{}:{}&0.178153850905&{}:{}&0.643692298190&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.213849483194&{}:{}&-0.213849483194&{}:{}&1.427698966388&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{7c}}\)
\(P^-_{\mathbf{7c}}\)
\(P^+_{\mathbf{7c}}\)
\(Q_{\mathbf{7c}}\)
\(I^\prime_{\mathbf{7c}}\)
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&-1.207106781187&{}:{}&1.707106781187&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.207106781187&{}:{}&0.500000000000&{}:{}&1.707106781187&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.066454497351&{}:{}&-0.066454497351&{}:{}&1.132908994701&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.125973796069&{}:{}&-0.036896870617&{}:{}&1.162870666686&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.036896870617&{}:{}&-0.125973796069&{}:{}&1.162870666686&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.353553390593&{}:{}&-0.353553390593&{}:{}&1.707106781187&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.060745576363&{}:{}&0.025161641584&{}:{}&1.035583934779&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.025161641584&{}:{}&-0.060745576363&{}:{}&1.035583934779&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.184699031259&{}:{}&0.184699031259&{}:{}&0.630601937482&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.216772751325&{}:{}&0.783227248675&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.216772751325&{}:{}&0.000000000000&{}:{}&0.783227248675&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.707106781187&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.707106781187&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.038928143267&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&-1.207106781187&{}:{}&1.707106781187&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.207106781187&{}:{}&0.500000000000&{}:{}&1.707106781187&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.066454497351&{}:{}&-0.066454497351&{}:{}&1.132908994701&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.033879030723&{}:{}&-0.033879030723&{}:{}&1.067758061446&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.125973796069&{}:{}&-0.036896870617&{}:{}&1.162870666686&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.036896870617&{}:{}&-0.125973796069&{}:{}&1.162870666686&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.353553390593&{}:{}&-0.353553390593&{}:{}&1.707106781187&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&0.023171078678&{}:{}&0.023171078678&{}:{}&0.953657842645&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.060745576363&{}:{}&0.025161641584&{}:{}&1.035583934779&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.025161641584&{}:{}&-0.060745576363&{}:{}&1.035583934779&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.184699031259&{}:{}&0.184699031259&{}:{}&0.630601937482&. \end{alignedat} \]
7c (332)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.094930300373&{}:{}&-0.094930300373&{}:{}&1.189860600745&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&-1.207106781187&{}:{}&1.707106781187&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-1.207106781187&{}:{}&0.500000000000&{}:{}&1.707106781187&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.066454497351&{}:{}&-0.066454497351&{}:{}&1.132908994701&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.125973796069&{}:{}&-0.036896870617&{}:{}&1.162870666686&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.036896870617&{}:{}&-0.125973796069&{}:{}&1.162870666686&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.353553390593&{}:{}&-0.353553390593&{}:{}&1.707106781187&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.178153850905&{}:{}&0.178153850905&{}:{}&0.643692298190&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.216772751325&{}:{}&0.783227248675&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.216772751325&{}:{}&0.000000000000&{}:{}&0.783227248675&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.213849483194&{}:{}&-0.213849483194&{}:{}&1.427698966388&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.125973796069&{}:{}&-0.036896870617&{}:{}&1.162870666686&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.036896870617&{}:{}&-0.125973796069&{}:{}&1.162870666686&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.353553390593&{}:{}&-0.353553390593&{}:{}&1.707106781187&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.216772751325&{}:{}&0.783227248675&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.216772751325&{}:{}&0.000000000000&{}:{}&0.783227248675&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)