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 > 3c (132)

3c(132)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.039215763225&{}:{}&-0.196738938710&{}:{}&1.235954701934&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.101789814961&{}:{}&-0.043014263180&{}:{}&0.941224448219&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.207377550454&{}:{}&-0.380069389256&{}:{}&1.587446939711&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.421354743656&{}:{}&-0.943762493583&{}:{}&1.522407749927&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.291005820742&{}:{}&-0.651803220840&{}:{}&1.942809041582&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{3c}}\)
\(P^-_{\mathbf{3c}}\)
\(P^+_{\mathbf{3c}}\)
\(Q_{\mathbf{3c}}\)
\(I^\prime_{\mathbf{3c}}\)
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.640652283836&{}:{}&-0.867542129971&{}:{}&1.226889846135&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-9.789312119522&{}:{}&-3.054865846209&{}:{}&13.844177965731&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.207106781187&{}:{}&-0.207106781187&{}:{}&1.414213562373&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.405631845412&{}:{}&-0.266106931081&{}:{}&1.671738776494&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.043726304683&{}:{}&-0.334386153871&{}:{}&1.378112458555&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.208419433030&{}:{}&-1.045605508834&{}:{}&2.254024941865&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.181278769177&{}:{}&-0.056570095410&{}:{}&1.237848864587&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.112453171914&{}:{}&-0.152278961217&{}:{}&1.039825789304&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.401206134927&{}:{}&-0.169541385685&{}:{}&0.768335250757&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-1.630986313698&{}:{}&2.630986313698&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.216772751325&{}:{}&0.000000000000&{}:{}&0.783227248675&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.806562964876&{}:{}&1.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.508194413807&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-5.734446273312&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.121320343560&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.640652283836&{}:{}&-0.867542129971&{}:{}&1.226889846135&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-9.789312119522&{}:{}&-3.054865846209&{}:{}&13.844177965731&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.207106781187&{}:{}&-0.207106781187&{}:{}&1.414213562373&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.039215763225&{}:{}&-0.196738938710&{}:{}&1.235954701934&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.405631845412&{}:{}&-0.266106931081&{}:{}&1.671738776494&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.043726304683&{}:{}&-0.334386153871&{}:{}&1.378112458555&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.208419433030&{}:{}&-1.045605508834&{}:{}&2.254024941865&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.101789814961&{}:{}&-0.043014263180&{}:{}&0.941224448219&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.181278769177&{}:{}&-0.056570095410&{}:{}&1.237848864587&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.112453171914&{}:{}&-0.152278961217&{}:{}&1.039825789304&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.401206134927&{}:{}&-0.169541385685&{}:{}&0.768335250757&. \end{alignedat} \]
3c (132)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.207377550454&{}:{}&-0.380069389256&{}:{}&1.587446939711&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.640652283836&{}:{}&-0.867542129971&{}:{}&1.226889846135&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-9.789312119522&{}:{}&-3.054865846209&{}:{}&13.844177965731&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.207106781187&{}:{}&-0.207106781187&{}:{}&1.414213562373&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.405631845412&{}:{}&-0.266106931081&{}:{}&1.671738776494&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.043726304683&{}:{}&-0.334386153871&{}:{}&1.378112458555&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.208419433030&{}:{}&-1.045605508834&{}:{}&2.254024941865&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.421354743656&{}:{}&-0.943762493583&{}:{}&1.522407749927&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-1.630986313698&{}:{}&2.630986313698&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.216772751325&{}:{}&0.000000000000&{}:{}&0.783227248675&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.806562964876&{}:{}&1.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.291005820742&{}:{}&-0.651803220840&{}:{}&1.942809041582&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.405631845412&{}:{}&-0.266106931081&{}:{}&1.671738776494&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.043726304683&{}:{}&-0.334386153871&{}:{}&1.378112458555&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.208419433030&{}:{}&-1.045605508834&{}:{}&2.254024941865&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-1.630986313698&{}:{}&2.630986313698&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.216772751325&{}:{}&0.000000000000&{}:{}&0.783227248675&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.806562964876&{}:{}&1.806562964876&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)