Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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1c(112)

Malfatti circles

1c (112)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&, \\ P_{\mathbf{1c}}&{}\approx{}&-0.165469293634&{}:{}&-0.337570405823&{}:{}&1.503039699457&, \\ P^-_{\mathbf{1c}}&{}\approx{}&0.149186857990&{}:{}&0.104065420702&{}:{}&0.746747721308&, \\ P^+_{\mathbf{1c}}&{}\approx{}&-0.671232574146&{}:{}&-1.047434755406&{}:{}&2.718667329552&, \\ Q_{\mathbf{1c}}&{}\approx{}&2.346612327186&{}:{}&2.736649604236&{}:{}&-4.083261931422&, \\ I^\prime_{\mathbf{1c}}&{}\approx{}&-0.805390790266&{}:{}&-1.328310188544&{}:{}&3.133700978809&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{1c}}\)
\(P^-_{\mathbf{1c}}\)
\(P^+_{\mathbf{1c}}\)
\(Q_{\mathbf{1c}}\)
\(I^\prime_{\mathbf{1c}}\)
1c (112)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1c}}&{}\approx{}&0.612295904741&{}:{}&-1.550816381038&{}:{}&1.938520476297&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.967232496600&{}:{}&0.355178335600&{}:{}&1.612054161000&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-3.314755078384&{}:{}&-4.419673437846&{}:{}&8.734428516230&, \\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.497500295804&{}:{}&-0.723384299357&{}:{}&3.220884595161&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.407743518916&{}:{}&-2.295993184536&{}:{}&3.703736703452&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.250085107259&{}:{}&-0.510193337347&{}:{}&1.760278444606&, \\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.499444201438&{}:{}&0.183401365044&{}:{}&1.316042836394&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.287963415461&{}:{}&-0.729350594018&{}:{}&1.441387178556&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.205611668514&{}:{}&0.143424595662&{}:{}&0.650963735825&, \\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-2.032247551123&{}:{}&3.032247551123&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.351229586822&{}:{}&0.000000000000&{}:{}&2.351229586822&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.461635138784&{}:{}&0.538364861216&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1c}}}{B^\prime_{\mathbf{1c}}}{C^\prime_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1c}}}{B^{\prime\prime\prime}_{\mathbf{1c}}}{C^{\prime\prime\prime}_{\mathbf{1c}}}\)
\(\triangle{A^*_{\mathbf{1c}}}{B^*_{\mathbf{1c}}}{C^*_{\mathbf{1c}}}\)
1c (112)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.775408190519&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1c}}}}&{}\approx{}&-0.644821664400&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1c}}}}&{}\approx{}&-2.209836718923&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.612295904741&{}:{}&-1.550816381038&{}:{}&1.938520476297&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.967232496600&{}:{}&0.355178335600&{}:{}&1.612054161000&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-3.314755078384&{}:{}&-4.419673437846&{}:{}&8.734428516230&. \end{alignedat} \]
1c (112)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1c}}&{}\approx{}&-0.165469293634&{}:{}&-0.337570405823&{}:{}&1.503039699457&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.497500295804&{}:{}&-0.723384299357&{}:{}&3.220884595161&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.407743518916&{}:{}&-2.295993184536&{}:{}&3.703736703452&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.250085107259&{}:{}&-0.510193337347&{}:{}&1.760278444606&. \end{alignedat} \]
1c (112)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1c}}&{}\approx{}&0.149186857990&{}:{}&0.104065420702&{}:{}&0.746747721308&,\\ A^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.499444201438&{}:{}&0.183401365044&{}:{}&1.316042836394&,\\B^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.287963415461&{}:{}&-0.729350594018&{}:{}&1.441387178556&,\\C^{\prime\prime\prime}_{\mathbf{1c}}&{}\approx{}&0.205611668514&{}:{}&0.143424595662&{}:{}&0.650963735825&. \end{alignedat} \]
1c (112)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1c}}&{}\approx{}&-0.671232574146&{}:{}&-1.047434755406&{}:{}&2.718667329552&,\\ A^\prime_{\mathbf{1c}}&{}\approx{}&0.612295904741&{}:{}&-1.550816381038&{}:{}&1.938520476297&,\\B^\prime_{\mathbf{1c}}&{}\approx{}&-0.967232496600&{}:{}&0.355178335600&{}:{}&1.612054161000&,\\C^\prime_{\mathbf{1c}}&{}\approx{}&-3.314755078384&{}:{}&-4.419673437846&{}:{}&8.734428516230&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.497500295804&{}:{}&-0.723384299357&{}:{}&3.220884595161&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.407743518916&{}:{}&-2.295993184536&{}:{}&3.703736703452&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.250085107259&{}:{}&-0.510193337347&{}:{}&1.760278444606&, \end{alignedat} \]
1c (112)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1c}}&{}\approx{}&2.346612327186&{}:{}&2.736649604236&{}:{}&-4.083261931422&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-2.032247551123&{}:{}&3.032247551123&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.351229586822&{}:{}&0.000000000000&{}:{}&2.351229586822&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.461635138784&{}:{}&0.538364861216&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1c}}&{}\approx{}&-0.805390790266&{}:{}&-1.328310188544&{}:{}&3.133700978809&,\\ A^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-1.497500295804&{}:{}&-0.723384299357&{}:{}&3.220884595161&,\\B^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.407743518916&{}:{}&-2.295993184536&{}:{}&3.703736703452&,\\C^{\prime\prime}_{\mathbf{1c}}&{}\approx{}&-0.250085107259&{}:{}&-0.510193337347&{}:{}&1.760278444606&,\\ A^*_{\mathbf{1c}}&{}\approx{}&0.000000000000&{}:{}&-2.032247551123&{}:{}&3.032247551123&,\\B^*_{\mathbf{1c}}&{}\approx{}&-1.351229586822&{}:{}&0.000000000000&{}:{}&2.351229586822&,\\C^*_{\mathbf{1c}}&{}\approx{}&0.461635138784&{}:{}&0.538364861216&{}:{}&0.000000000000&. \end{alignedat} \]
1c (112)