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 > 2b (121)

2b(121)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.017098243047&{}:{}&1.124512942078&{}:{}&-0.107414699030&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.164533597151&{}:{}&0.665139629621&{}:{}&0.170326773228&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.381530562809&{}:{}&2.046215225733&{}:{}&-0.664684662924&, \\ Q_{\mathbf{2b}}&{}\approx{}&-1.340419821139&{}:{}&4.502685791374&{}:{}&-2.162265970234&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.242185659742&{}:{}&1.964060720257&{}:{}&-0.721875060515&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{2b}}\)
\(P^-_{\mathbf{2b}}\)
\(P^+_{\mathbf{2b}}\)
\(Q_{\mathbf{2b}}\)
\(I^\prime_{\mathbf{2b}}\)
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&1.806562964876&{}:{}&1.947215248712&{}:{}&-2.753778213589&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.748302881333&{}:{}&2.806562964876&{}:{}&-1.058260083544&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-1.041196100146&{}:{}&1.041196100146&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.870780089663&{}:{}&2.068351250224&{}:{}&-0.197571160561&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.049004836454&{}:{}&1.356863354089&{}:{}&-0.307858517635&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.039046485382&{}:{}&2.567999415663&{}:{}&-1.528952930282&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.269468803895&{}:{}&1.010661837698&{}:{}&0.258806966197&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.321226729290&{}:{}&0.346236249568&{}:{}&0.332537021143&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.284231621722&{}:{}&1.149028033617&{}:{}&-0.433259655339&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.923879532511&{}:{}&-0.923879532511&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.382683432365&{}:{}&0.000000000000&{}:{}&0.617316567635&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-2.753778213589&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.058260083544&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.472473645917&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.707106781187&{}:{}&-0.707106781187&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&1.806562964876&{}:{}&1.947215248712&{}:{}&-2.753778213589&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.748302881333&{}:{}&2.806562964876&{}:{}&-1.058260083544&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-1.041196100146&{}:{}&1.041196100146&{}:{}&1.000000000000&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.017098243047&{}:{}&1.124512942078&{}:{}&-0.107414699030&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.870780089663&{}:{}&2.068351250224&{}:{}&-0.197571160561&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.049004836454&{}:{}&1.356863354089&{}:{}&-0.307858517635&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.039046485382&{}:{}&2.567999415663&{}:{}&-1.528952930282&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.164533597151&{}:{}&0.665139629621&{}:{}&0.170326773228&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.269468803895&{}:{}&1.010661837698&{}:{}&0.258806966197&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.321226729290&{}:{}&0.346236249568&{}:{}&0.332537021143&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.284231621722&{}:{}&1.149028033617&{}:{}&-0.433259655339&. \end{alignedat} \]
2b (121)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.381530562809&{}:{}&2.046215225733&{}:{}&-0.664684662924&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&1.806562964876&{}:{}&1.947215248712&{}:{}&-2.753778213589&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.748302881333&{}:{}&2.806562964876&{}:{}&-1.058260083544&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-1.041196100146&{}:{}&1.041196100146&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.870780089663&{}:{}&2.068351250224&{}:{}&-0.197571160561&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.049004836454&{}:{}&1.356863354089&{}:{}&-0.307858517635&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.039046485382&{}:{}&2.567999415663&{}:{}&-1.528952930282&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&-1.340419821139&{}:{}&4.502685791374&{}:{}&-2.162265970234&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.923879532511&{}:{}&-0.923879532511&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.382683432365&{}:{}&0.000000000000&{}:{}&0.617316567635&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.242185659742&{}:{}&1.964060720257&{}:{}&-0.721875060515&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.870780089663&{}:{}&2.068351250224&{}:{}&-0.197571160561&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.049004836454&{}:{}&1.356863354089&{}:{}&-0.307858517635&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.039046485382&{}:{}&2.567999415663&{}:{}&-1.528952930282&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.923879532511&{}:{}&-0.923879532511&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.382683432365&{}:{}&0.000000000000&{}:{}&0.617316567635&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.423879532511&{}:{}&1.423879532511&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)