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 > 1c (112)

1c(112)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.233524451931&{}:{}&-0.233524451931&{}:{}&1.467048903863&, \\ P^-_{\mathbf{1c}}&{}\approx{}&0.133865788615&{}:{}&0.133865788615&{}:{}&0.732268422770&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-0.824782511313&{}:{}&-0.824782511313&{}:{}&2.649565022626&, \\ Q_{\mathbf{1c}}&{}\approx{}&2.584820428417&{}:{}&2.584820428417&{}:{}&-4.169640856834&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-1.019713031423&{}:{}&-1.019713031423&{}:{}&3.039426062845&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{1c}}\)
\(P^-_{\mathbf{1c}}\)
\(P^+_{\mathbf{1c}}\)
\(Q_{\mathbf{1c}}\)
\(I^\prime_{\mathbf{1c}}\)
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&-1.207106781187&{}:{}&1.707106781187&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.207106781187&{}:{}&0.500000000000&{}:{}&1.707106781187&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-3.761972627396&{}:{}&-3.761972627396&{}:{}&8.523945254791&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.827744053083&{}:{}&-0.535333838875&{}:{}&3.363077891958&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.535333838875&{}:{}&-1.827744053083&{}:{}&3.363077891958&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.353553390593&{}:{}&-0.353553390593&{}:{}&1.707106781187&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.595226916389&{}:{}&0.246551061458&{}:{}&1.348675854931&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.246551061458&{}:{}&-0.595226916389&{}:{}&1.348675854931&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.184699031259&{}:{}&0.184699031259&{}:{}&0.630601937482&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-1.630986313698&{}:{}&2.630986313698&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.630986313698&{}:{}&0.000000000000&{}:{}&2.630986313698&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.707106781187&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.707106781187&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-2.203712543852&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.707106781187&{}:{}&1.707106781187&{}:{}&-2.414213562373&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&-1.207106781187&{}:{}&1.707106781187&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.207106781187&{}:{}&0.500000000000&{}:{}&1.707106781187&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-3.761972627396&{}:{}&-3.761972627396&{}:{}&8.523945254791&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.233524451931&{}:{}&-0.233524451931&{}:{}&1.467048903863&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.827744053083&{}:{}&-0.535333838875&{}:{}&3.363077891958&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.535333838875&{}:{}&-1.827744053083&{}:{}&3.363077891958&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.353553390593&{}:{}&-0.353553390593&{}:{}&1.707106781187&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&0.133865788615&{}:{}&0.133865788615&{}:{}&0.732268422770&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.595226916389&{}:{}&0.246551061458&{}:{}&1.348675854931&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.246551061458&{}:{}&-0.595226916389&{}:{}&1.348675854931&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.184699031259&{}:{}&0.184699031259&{}:{}&0.630601937482&. \end{alignedat} \]
1c (112)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-0.824782511313&{}:{}&-0.824782511313&{}:{}&2.649565022626&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&-1.207106781187&{}:{}&1.707106781187&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-1.207106781187&{}:{}&0.500000000000&{}:{}&1.707106781187&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-3.761972627396&{}:{}&-3.761972627396&{}:{}&8.523945254791&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.827744053083&{}:{}&-0.535333838875&{}:{}&3.363077891958&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.535333838875&{}:{}&-1.827744053083&{}:{}&3.363077891958&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.353553390593&{}:{}&-0.353553390593&{}:{}&1.707106781187&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&2.584820428417&{}:{}&2.584820428417&{}:{}&-4.169640856834&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-1.630986313698&{}:{}&2.630986313698&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.630986313698&{}:{}&0.000000000000&{}:{}&2.630986313698&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-1.019713031423&{}:{}&-1.019713031423&{}:{}&3.039426062845&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.827744053083&{}:{}&-0.535333838875&{}:{}&3.363077891958&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.535333838875&{}:{}&-1.827744053083&{}:{}&3.363077891958&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.353553390593&{}:{}&-0.353553390593&{}:{}&1.707106781187&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-1.630986313698&{}:{}&2.630986313698&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.630986313698&{}:{}&0.000000000000&{}:{}&2.630986313698&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.500000000000&{}:{}&0.500000000000&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)