Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


[Top] > The Smallest Pythagorean Triangle > 2b (121)

2b(121)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.021811136794&{}:{}&1.219932514912&{}:{}&-0.198121378118&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.165976642650&{}:{}&0.679805302934&{}:{}&0.154218054416&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.387595300582&{}:{}&2.272024302558&{}:{}&-0.884429001976&, \\ Q_{\mathbf{2b}}&{}\approx{}&-3.193680423844&{}:{}&9.750897329564&{}:{}&-5.557216905720&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.267036670875&{}:{}&2.306051627355&{}:{}&-1.039014956480&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{2b}}\)
\(P^-_{\mathbf{2b}}\)
\(P^+_{\mathbf{2b}}\)
\(Q_{\mathbf{2b}}\)
\(I^\prime_{\mathbf{2b}}\)
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&1.578480113194&{}:{}&2.313920452774&{}:{}&-2.892400565968&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.944570142487&{}:{}&3.518853713298&{}:{}&-1.574283570811&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.848570869931&{}:{}&1.131427826575&{}:{}&0.717143043356&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.894000752836&{}:{}&2.261233038526&{}:{}&-0.367232285690&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.051544619992&{}:{}&1.519749979671&{}:{}&-0.468205359679&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.055320658033&{}:{}&3.094174784169&{}:{}&-2.038854126136&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.280075335746&{}:{}&1.043378454244&{}:{}&0.236696881502&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.294544570142&{}:{}&0.431777821848&{}:{}&0.273677608011&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.306429532844&{}:{}&1.255070701983&{}:{}&-0.561500234827&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.325140769936&{}:{}&-1.325140769936&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.364954621631&{}:{}&0.000000000000&{}:{}&0.635045378369&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.487048159267&{}:{}&1.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-2.313920452774&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.259426856649&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.131427826575&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&1.578480113194&{}:{}&2.313920452774&{}:{}&-2.892400565968&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.944570142487&{}:{}&3.518853713298&{}:{}&-1.574283570811&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.848570869931&{}:{}&1.131427826575&{}:{}&0.717143043356&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.021811136794&{}:{}&1.219932514912&{}:{}&-0.198121378118&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.894000752836&{}:{}&2.261233038526&{}:{}&-0.367232285690&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.051544619992&{}:{}&1.519749979671&{}:{}&-0.468205359679&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.055320658033&{}:{}&3.094174784169&{}:{}&-2.038854126136&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.165976642650&{}:{}&0.679805302934&{}:{}&0.154218054416&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.280075335746&{}:{}&1.043378454244&{}:{}&0.236696881502&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.294544570142&{}:{}&0.431777821848&{}:{}&0.273677608011&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.306429532844&{}:{}&1.255070701983&{}:{}&-0.561500234827&. \end{alignedat} \]
2b (121)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.387595300582&{}:{}&2.272024302558&{}:{}&-0.884429001976&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&1.578480113194&{}:{}&2.313920452774&{}:{}&-2.892400565968&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.944570142487&{}:{}&3.518853713298&{}:{}&-1.574283570811&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.848570869931&{}:{}&1.131427826575&{}:{}&0.717143043356&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.894000752836&{}:{}&2.261233038526&{}:{}&-0.367232285690&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.051544619992&{}:{}&1.519749979671&{}:{}&-0.468205359679&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.055320658033&{}:{}&3.094174784169&{}:{}&-2.038854126136&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&-3.193680423844&{}:{}&9.750897329564&{}:{}&-5.557216905720&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.325140769936&{}:{}&-1.325140769936&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.364954621631&{}:{}&0.000000000000&{}:{}&0.635045378369&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.487048159267&{}:{}&1.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.267036670875&{}:{}&2.306051627355&{}:{}&-1.039014956480&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.894000752836&{}:{}&2.261233038526&{}:{}&-0.367232285690&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.051544619992&{}:{}&1.519749979671&{}:{}&-0.468205359679&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.055320658033&{}:{}&3.094174784169&{}:{}&-2.038854126136&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.325140769936&{}:{}&-1.325140769936&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.364954621631&{}:{}&0.000000000000&{}:{}&0.635045378369&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.487048159267&{}:{}&1.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)