Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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3b(123)

Malfatti circles

3b (123)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&, \\ P_{\mathbf{3b}}&{}\approx{}&-0.005793987199&{}:{}&1.200809762299&{}:{}&-0.195015775099&, \\ P^-_{\mathbf{3b}}&{}\approx{}&0.100926923260&{}:{}&0.890047272047&{}:{}&0.009025804694&, \\ P^+_{\mathbf{3b}}&{}\approx{}&-0.154514739965&{}:{}&1.633872329725&{}:{}&-0.479357589761&, \\ Q_{\mathbf{3b}}&{}\approx{}&0.283471782768&{}:{}&1.666028970695&{}:{}&-0.949500753464&, \\ I^\prime_{\mathbf{3b}}&{}\approx{}&-0.122948098990&{}:{}&2.043805322007&{}:{}&-0.920857223017&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{3b}}\)
\(P^-_{\mathbf{3b}}\)
\(P^+_{\mathbf{3b}}\)
\(Q_{\mathbf{3b}}\)
\(I^\prime_{\mathbf{3b}}\)
3b (123)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3b}}&{}\approx{}&1.537910698151&{}:{}&2.151642792606&{}:{}&-2.689553490757&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.174285434966&{}:{}&1.464761159908&{}:{}&-0.290475724943&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.389140284973&{}:{}&3.185520379965&{}:{}&0.203619905009&, \\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.324871792088&{}:{}&1.581754317465&{}:{}&-0.256882525377&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.045254491075&{}:{}&2.568443851024&{}:{}&-1.523189359949&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.007390623871&{}:{}&1.531714342500&{}:{}&-0.524323718629&, \\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.133518335437&{}:{}&1.122138932164&{}:{}&0.011379403273&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.401848849627&{}:{}&0.562214166307&{}:{}&0.035936984066&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.123917366050&{}:{}&1.092793776428&{}:{}&-0.216711142478&, \\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.325140769936&{}:{}&-1.325140769936&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.425614793411&{}:{}&0.000000000000&{}:{}&1.425614793411&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.145407372767&{}:{}&0.854592627233&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3b}}}{B^\prime_{\mathbf{3b}}}{C^\prime_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3b}}}{B^{\prime\prime\prime}_{\mathbf{3b}}}{C^{\prime\prime\prime}_{\mathbf{3b}}}\)
\(\triangle{A^*_{\mathbf{3b}}}{B^*_{\mathbf{3b}}}{C^*_{\mathbf{3b}}}\)
3b (123)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3b}}}}&{}\approx{}&-2.151642792606&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3b}}}}&{}\approx{}&-0.232380579954&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3b}}}}&{}\approx{}&-3.185520379965&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.537910698151&{}:{}&2.151642792606&{}:{}&-2.689553490757&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.174285434966&{}:{}&1.464761159908&{}:{}&-0.290475724943&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.389140284973&{}:{}&3.185520379965&{}:{}&0.203619905009&. \end{alignedat} \]
3b (123)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3b}}&{}\approx{}&-0.005793987199&{}:{}&1.200809762299&{}:{}&-0.195015775099&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.324871792088&{}:{}&1.581754317465&{}:{}&-0.256882525377&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.045254491075&{}:{}&2.568443851024&{}:{}&-1.523189359949&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.007390623871&{}:{}&1.531714342500&{}:{}&-0.524323718629&. \end{alignedat} \]
3b (123)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3b}}&{}\approx{}&0.100926923260&{}:{}&0.890047272047&{}:{}&0.009025804694&,\\ A^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.133518335437&{}:{}&1.122138932164&{}:{}&0.011379403273&,\\B^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.401848849627&{}:{}&0.562214166307&{}:{}&0.035936984066&,\\C^{\prime\prime\prime}_{\mathbf{3b}}&{}\approx{}&0.123917366050&{}:{}&1.092793776428&{}:{}&-0.216711142478&. \end{alignedat} \]
3b (123)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3b}}&{}\approx{}&-0.154514739965&{}:{}&1.633872329725&{}:{}&-0.479357589761&,\\ A^\prime_{\mathbf{3b}}&{}\approx{}&1.537910698151&{}:{}&2.151642792606&{}:{}&-2.689553490757&,\\B^\prime_{\mathbf{3b}}&{}\approx{}&-0.174285434966&{}:{}&1.464761159908&{}:{}&-0.290475724943&,\\C^\prime_{\mathbf{3b}}&{}\approx{}&-2.389140284973&{}:{}&3.185520379965&{}:{}&0.203619905009&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.324871792088&{}:{}&1.581754317465&{}:{}&-0.256882525377&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.045254491075&{}:{}&2.568443851024&{}:{}&-1.523189359949&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.007390623871&{}:{}&1.531714342500&{}:{}&-0.524323718629&, \end{alignedat} \]
3b (123)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3b}}&{}\approx{}&0.283471782768&{}:{}&1.666028970695&{}:{}&-0.949500753464&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.325140769936&{}:{}&-1.325140769936&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.425614793411&{}:{}&0.000000000000&{}:{}&1.425614793411&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.145407372767&{}:{}&0.854592627233&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3b}}&{}\approx{}&-0.122948098990&{}:{}&2.043805322007&{}:{}&-0.920857223017&,\\ A^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.324871792088&{}:{}&1.581754317465&{}:{}&-0.256882525377&,\\B^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.045254491075&{}:{}&2.568443851024&{}:{}&-1.523189359949&,\\C^{\prime\prime}_{\mathbf{3b}}&{}\approx{}&-0.007390623871&{}:{}&1.531714342500&{}:{}&-0.524323718629&,\\ A^*_{\mathbf{3b}}&{}\approx{}&0.000000000000&{}:{}&2.325140769936&{}:{}&-1.325140769936&,\\B^*_{\mathbf{3b}}&{}\approx{}&-0.425614793411&{}:{}&0.000000000000&{}:{}&1.425614793411&,\\C^*_{\mathbf{3b}}&{}\approx{}&0.145407372767&{}:{}&0.854592627233&{}:{}&0.000000000000&. \end{alignedat} \]
3b (123)