Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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3c(132)

Malfatti circles

3c (132)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&, \\ P_{\mathbf{3c}}&{}\approx{}&-0.039842346058&{}:{}&-0.301183398540&{}:{}&1.341025744598&, \\ P^-_{\mathbf{3c}}&{}\approx{}&0.110208259647&{}:{}&-0.076943583553&{}:{}&0.966735323906&, \\ P^+_{\mathbf{3c}}&{}\approx{}&-0.226215347855&{}:{}&-0.579704416467&{}:{}&1.805919764322&, \\ Q_{\mathbf{3c}}&{}\approx{}&0.475140607557&{}:{}&-1.066644214975&{}:{}&1.591503607418&, \\ I^\prime_{\mathbf{3c}}&{}\approx{}&-0.301656096168&{}:{}&-0.957689085290&{}:{}&2.259345181458&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{3c}}\)
\(P^-_{\mathbf{3c}}\)
\(P^+_{\mathbf{3c}}\)
\(Q_{\mathbf{3c}}\)
\(I^\prime_{\mathbf{3c}}\)
3c (132)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3c}}&{}\approx{}&0.693434734825&{}:{}&-1.226261060701&{}:{}&1.532826325877&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-7.129510156769&{}:{}&-3.753006771179&{}:{}&11.882516927948&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.233616248300&{}:{}&-0.311488331067&{}:{}&1.545104579367&, \\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.452408209071&{}:{}&-0.420680348452&{}:{}&1.873088557524&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.045831707403&{}:{}&-0.496785774450&{}:{}&1.542617481854&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.200408653961&{}:{}&-1.514964992501&{}:{}&2.715373646461&, \\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.196562014835&{}:{}&-0.103471144077&{}:{}&1.300033158912&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.124945207587&{}:{}&-0.220951497077&{}:{}&1.096006289491&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.398038687510&{}:{}&-0.277896802906&{}:{}&0.879858115396&, \\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-2.032247551123&{}:{}&3.032247551123&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.229909243262&{}:{}&0.000000000000&{}:{}&0.770090756738&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.803275925284&{}:{}&1.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3c}}}{B^\prime_{\mathbf{3c}}}{C^\prime_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3c}}}{B^{\prime\prime\prime}_{\mathbf{3c}}}{C^{\prime\prime\prime}_{\mathbf{3c}}}\)
\(\triangle{A^*_{\mathbf{3c}}}{B^*_{\mathbf{3c}}}{C^*_{\mathbf{3c}}}\)
3c (132)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.613130530351&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3c}}}}&{}\approx{}&-4.753006771179&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3c}}}}&{}\approx{}&-0.155744165533&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.500000000000&{}:{}&2.000000000000&{}:{}&-2.500000000000&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.693434734825&{}:{}&-1.226261060701&{}:{}&1.532826325877&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-7.129510156769&{}:{}&-3.753006771179&{}:{}&11.882516927948&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.233616248300&{}:{}&-0.311488331067&{}:{}&1.545104579367&. \end{alignedat} \]
3c (132)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3c}}&{}\approx{}&-0.039842346058&{}:{}&-0.301183398540&{}:{}&1.341025744598&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.452408209071&{}:{}&-0.420680348452&{}:{}&1.873088557524&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.045831707403&{}:{}&-0.496785774450&{}:{}&1.542617481854&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.200408653961&{}:{}&-1.514964992501&{}:{}&2.715373646461&. \end{alignedat} \]
3c (132)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3c}}&{}\approx{}&0.110208259647&{}:{}&-0.076943583553&{}:{}&0.966735323906&,\\ A^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.196562014835&{}:{}&-0.103471144077&{}:{}&1.300033158912&,\\B^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.124945207587&{}:{}&-0.220951497077&{}:{}&1.096006289491&,\\C^{\prime\prime\prime}_{\mathbf{3c}}&{}\approx{}&0.398038687510&{}:{}&-0.277896802906&{}:{}&0.879858115396&. \end{alignedat} \]
3c (132)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3c}}&{}\approx{}&-0.226215347855&{}:{}&-0.579704416467&{}:{}&1.805919764322&,\\ A^\prime_{\mathbf{3c}}&{}\approx{}&0.693434734825&{}:{}&-1.226261060701&{}:{}&1.532826325877&,\\B^\prime_{\mathbf{3c}}&{}\approx{}&-7.129510156769&{}:{}&-3.753006771179&{}:{}&11.882516927948&,\\C^\prime_{\mathbf{3c}}&{}\approx{}&-0.233616248300&{}:{}&-0.311488331067&{}:{}&1.545104579367&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.452408209071&{}:{}&-0.420680348452&{}:{}&1.873088557524&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.045831707403&{}:{}&-0.496785774450&{}:{}&1.542617481854&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.200408653961&{}:{}&-1.514964992501&{}:{}&2.715373646461&, \end{alignedat} \]
3c (132)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3c}}&{}\approx{}&0.475140607557&{}:{}&-1.066644214975&{}:{}&1.591503607418&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-2.032247551123&{}:{}&3.032247551123&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.229909243262&{}:{}&0.000000000000&{}:{}&0.770090756738&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.803275925284&{}:{}&1.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3c}}&{}\approx{}&-0.301656096168&{}:{}&-0.957689085290&{}:{}&2.259345181458&,\\ A^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.452408209071&{}:{}&-0.420680348452&{}:{}&1.873088557524&,\\B^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.045831707403&{}:{}&-0.496785774450&{}:{}&1.542617481854&,\\C^{\prime\prime}_{\mathbf{3c}}&{}\approx{}&-0.200408653961&{}:{}&-1.514964992501&{}:{}&2.715373646461&,\\ A^*_{\mathbf{3c}}&{}\approx{}&0.000000000000&{}:{}&-2.032247551123&{}:{}&3.032247551123&,\\B^*_{\mathbf{3c}}&{}\approx{}&0.229909243262&{}:{}&0.000000000000&{}:{}&0.770090756738&,\\C^*_{\mathbf{3c}}&{}\approx{}&-0.803275925284&{}:{}&1.803275925284&{}:{}&0.000000000000&. \end{alignedat} \]
3c (132)