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 > 4a (211)

4a(211)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{4a}}&{}\approx{}&1.124512942078&{}:{}&-0.017098243047&{}:{}&-0.107414699030&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.665139629621&{}:{}&0.164533597151&{}:{}&0.170326773228&, \\ P^+_{\mathbf{4a}}&{}\approx{}&2.046215225733&{}:{}&-0.381530562809&{}:{}&-0.664684662924&, \\ Q_{\mathbf{4a}}&{}\approx{}&4.502685791374&{}:{}&-1.340419821139&{}:{}&-2.162265970234&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.964060720257&{}:{}&-0.242185659742&{}:{}&-0.721875060515&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&2.806562964876&{}:{}&-0.748302881333&{}:{}&-1.058260083544&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.947215248712&{}:{}&1.806562964876&{}:{}&-2.753778213589&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&1.041196100146&{}:{}&-1.041196100146&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.356863354089&{}:{}&-0.049004836454&{}:{}&-0.307858517635&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.068351250224&{}:{}&-0.870780089663&{}:{}&-0.197571160561&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.567999415663&{}:{}&-0.039046485382&{}:{}&-1.528952930282&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.346236249568&{}:{}&0.321226729290&{}:{}&0.332537021143&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.010661837698&{}:{}&-0.269468803895&{}:{}&0.258806966197&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.149028033617&{}:{}&0.284231621722&{}:{}&-0.433259655339&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.382683432365&{}:{}&0.617316567635&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.923879532511&{}:{}&0.000000000000&{}:{}&-0.923879532511&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.058260083544&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-2.753778213589&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.472473645917&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.707106781187&{}:{}&0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.806562964876&{}:{}&-0.748302881333&{}:{}&-1.058260083544&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.947215248712&{}:{}&1.806562964876&{}:{}&-2.753778213589&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&1.041196100146&{}:{}&-1.041196100146&{}:{}&1.000000000000&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.124512942078&{}:{}&-0.017098243047&{}:{}&-0.107414699030&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.356863354089&{}:{}&-0.049004836454&{}:{}&-0.307858517635&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.068351250224&{}:{}&-0.870780089663&{}:{}&-0.197571160561&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.567999415663&{}:{}&-0.039046485382&{}:{}&-1.528952930282&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.665139629621&{}:{}&0.164533597151&{}:{}&0.170326773228&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.346236249568&{}:{}&0.321226729290&{}:{}&0.332537021143&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.010661837698&{}:{}&-0.269468803895&{}:{}&0.258806966197&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.149028033617&{}:{}&0.284231621722&{}:{}&-0.433259655339&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&2.046215225733&{}:{}&-0.381530562809&{}:{}&-0.664684662924&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.806562964876&{}:{}&-0.748302881333&{}:{}&-1.058260083544&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&1.947215248712&{}:{}&1.806562964876&{}:{}&-2.753778213589&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&1.041196100146&{}:{}&-1.041196100146&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.356863354089&{}:{}&-0.049004836454&{}:{}&-0.307858517635&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.068351250224&{}:{}&-0.870780089663&{}:{}&-0.197571160561&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.567999415663&{}:{}&-0.039046485382&{}:{}&-1.528952930282&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&4.502685791374&{}:{}&-1.340419821139&{}:{}&-2.162265970234&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.382683432365&{}:{}&0.617316567635&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.923879532511&{}:{}&0.000000000000&{}:{}&-0.923879532511&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.964060720257&{}:{}&-0.242185659742&{}:{}&-0.721875060515&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.356863354089&{}:{}&-0.049004836454&{}:{}&-0.307858517635&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.068351250224&{}:{}&-0.870780089663&{}:{}&-0.197571160561&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.567999415663&{}:{}&-0.039046485382&{}:{}&-1.528952930282&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.382683432365&{}:{}&0.617316567635&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.923879532511&{}:{}&0.000000000000&{}:{}&-0.923879532511&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.423879532511&{}:{}&-0.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)