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 > 5c (312)

5c(312)

Malfatti circles

5c (312)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&, \\ P_{\mathbf{5c}}&{}\approx{}&-0.196738938710&{}:{}&-0.039215763225&{}:{}&1.235954701934&, \\ P^-_{\mathbf{5c}}&{}\approx{}&-0.043014263180&{}:{}&0.101789814961&{}:{}&0.941224448219&, \\ P^+_{\mathbf{5c}}&{}\approx{}&-0.380069389256&{}:{}&-0.207377550454&{}:{}&1.587446939711&, \\ Q_{\mathbf{5c}}&{}\approx{}&-0.943762493583&{}:{}&0.421354743656&{}:{}&1.522407749927&, \\ I^\prime_{\mathbf{5c}}&{}\approx{}&-0.651803220840&{}:{}&-0.291005820742&{}:{}&1.942809041582&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{5c}}\)
\(P^-_{\mathbf{5c}}\)
\(P^+_{\mathbf{5c}}\)
\(Q_{\mathbf{5c}}\)
\(I^\prime_{\mathbf{5c}}\)
5c (312)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5c}}&{}\approx{}&-3.054865846209&{}:{}&-9.789312119522&{}:{}&13.844177965731&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.867542129971&{}:{}&0.640652283836&{}:{}&1.226889846135&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.207106781187&{}:{}&-0.207106781187&{}:{}&1.414213562373&, \\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.334386153871&{}:{}&-0.043726304683&{}:{}&1.378112458555&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.266106931081&{}:{}&-0.405631845412&{}:{}&1.671738776494&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.045605508834&{}:{}&-0.208419433030&{}:{}&2.254024941865&, \\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.152278961217&{}:{}&0.112453171914&{}:{}&1.039825789304&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.056570095410&{}:{}&-0.181278769177&{}:{}&1.237848864587&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.169541385685&{}:{}&0.401206134927&{}:{}&0.768335250757&, \\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.216772751325&{}:{}&0.783227248675&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.630986313698&{}:{}&0.000000000000&{}:{}&2.630986313698&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.806562964876&{}:{}&-0.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5c}}}{B^\prime_{\mathbf{5c}}}{C^\prime_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^*_{\mathbf{5c}}}{B^*_{\mathbf{5c}}}{C^*_{\mathbf{5c}}}\)
5c (312)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5c}}}}&{}\approx{}&-5.734446273312&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.508194413807&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.121320343560&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-3.054865846209&{}:{}&-9.789312119522&{}:{}&13.844177965731&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.867542129971&{}:{}&0.640652283836&{}:{}&1.226889846135&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.207106781187&{}:{}&-0.207106781187&{}:{}&1.414213562373&. \end{alignedat} \]
5c (312)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5c}}&{}\approx{}&-0.196738938710&{}:{}&-0.039215763225&{}:{}&1.235954701934&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.334386153871&{}:{}&-0.043726304683&{}:{}&1.378112458555&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.266106931081&{}:{}&-0.405631845412&{}:{}&1.671738776494&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.045605508834&{}:{}&-0.208419433030&{}:{}&2.254024941865&. \end{alignedat} \]
5c (312)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5c}}&{}\approx{}&-0.043014263180&{}:{}&0.101789814961&{}:{}&0.941224448219&,\\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.152278961217&{}:{}&0.112453171914&{}:{}&1.039825789304&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.056570095410&{}:{}&-0.181278769177&{}:{}&1.237848864587&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.169541385685&{}:{}&0.401206134927&{}:{}&0.768335250757&. \end{alignedat} \]
5c (312)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5c}}&{}\approx{}&-0.380069389256&{}:{}&-0.207377550454&{}:{}&1.587446939711&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-3.054865846209&{}:{}&-9.789312119522&{}:{}&13.844177965731&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.867542129971&{}:{}&0.640652283836&{}:{}&1.226889846135&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.207106781187&{}:{}&-0.207106781187&{}:{}&1.414213562373&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.334386153871&{}:{}&-0.043726304683&{}:{}&1.378112458555&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.266106931081&{}:{}&-0.405631845412&{}:{}&1.671738776494&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.045605508834&{}:{}&-0.208419433030&{}:{}&2.254024941865&, \end{alignedat} \]
5c (312)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5c}}&{}\approx{}&-0.943762493583&{}:{}&0.421354743656&{}:{}&1.522407749927&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.216772751325&{}:{}&0.783227248675&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.630986313698&{}:{}&0.000000000000&{}:{}&2.630986313698&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.806562964876&{}:{}&-0.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5c}}&{}\approx{}&-0.651803220840&{}:{}&-0.291005820742&{}:{}&1.942809041582&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.334386153871&{}:{}&-0.043726304683&{}:{}&1.378112458555&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.266106931081&{}:{}&-0.405631845412&{}:{}&1.671738776494&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-1.045605508834&{}:{}&-0.208419433030&{}:{}&2.254024941865&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.216772751325&{}:{}&0.783227248675&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.630986313698&{}:{}&0.000000000000&{}:{}&2.630986313698&,\\C^*_{\mathbf{5c}}&{}\approx{}&1.806562964876&{}:{}&-0.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)