Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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6(220)

Malfatti circles

6 (220)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&, \\ P_{\mathbf{6}}&{}\approx{}&0.759568951087&{}:{}&0.239287085717&{}:{}&0.001143963197&, \\ P^-_{\mathbf{6}}&{}\approx{}&1.148008781039&{}:{}&0.167596476715&{}:{}&-0.315605257754&, \\ P^+_{\mathbf{6}}&{}\approx{}&0.605705924598&{}:{}&0.267684106880&{}:{}&0.126609968522&, \\ Q_{\mathbf{6}}&{}\approx{}&0.617207145668&{}:{}&0.424280042126&{}:{}&-0.041487187795&, \\ I^\prime_{\mathbf{6}}&{}\approx{}&0.588856682171&{}:{}&0.381640981690&{}:{}&0.029502336139&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{6}}\)
\(P^-_{\mathbf{6}}\)
\(P^+_{\mathbf{6}}\)
\(Q_{\mathbf{6}}\)
\(I^\prime_{\mathbf{6}}\)
6 (220)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6}}&{}\approx{}&0.667552905630&{}:{}&0.147754264164&{}:{}&0.184692830205&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.281999757518&{}:{}&0.248000646617&{}:{}&0.469999595864&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.476569900005&{}:{}&1.968759866674&{}:{}&-2.445329766679&, \\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.473557987287&{}:{}&0.523937218551&{}:{}&0.002504794162&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.724030539754&{}:{}&0.274879020346&{}:{}&0.001090439899&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.558785949906&{}:{}&0.176034395957&{}:{}&0.265179654137&, \\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&4.477325463352&{}:{}&3.937519733347&{}:{}&-7.414845196699&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.056611343681&{}:{}&0.233867353848&{}:{}&-0.290478697529&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.702908489980&{}:{}&0.102616799035&{}:{}&0.194474710985&, \\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.108380256645&{}:{}&-0.108380256645&,\\B^*_{\mathbf{6}}&{}\approx{}&1.072061402818&{}:{}&0.000000000000&{}:{}&-0.072061402818&,\\C^*_{\mathbf{6}}&{}\approx{}&0.592620968267&{}:{}&0.407379031733&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6}}}{B^\prime_{\mathbf{6}}}{C^\prime_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6}}}{B^{\prime\prime}_{\mathbf{6}}}{C^{\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6}}}{B^{\prime\prime\prime}_{\mathbf{6}}}{C^{\prime\prime\prime}_{\mathbf{6}}}\)
\(\triangle{A^*_{\mathbf{6}}}{B^*_{\mathbf{6}}}{C^*_{\mathbf{6}}}\)
6 (220)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6}}}}&{}\approx{}&0.443262792493&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6}}}}&{}\approx{}&1.127999030074&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6}}}}&{}\approx{}&5.906279600021&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.250000000000&{}:{}&0.333333333333&{}:{}&0.416666666667&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.667552905630&{}:{}&0.147754264164&{}:{}&0.184692830205&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.281999757518&{}:{}&0.248000646617&{}:{}&0.469999595864&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.476569900005&{}:{}&1.968759866674&{}:{}&-2.445329766679&. \end{alignedat} \]
6 (220)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6}}&{}\approx{}&0.759568951087&{}:{}&0.239287085717&{}:{}&0.001143963197&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.473557987287&{}:{}&0.523937218551&{}:{}&0.002504794162&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.724030539754&{}:{}&0.274879020346&{}:{}&0.001090439899&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.558785949906&{}:{}&0.176034395957&{}:{}&0.265179654137&. \end{alignedat} \]
6 (220)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6}}&{}\approx{}&1.148008781039&{}:{}&0.167596476715&{}:{}&-0.315605257754&,\\ A^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&4.477325463352&{}:{}&3.937519733347&{}:{}&-7.414845196699&,\\B^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&1.056611343681&{}:{}&0.233867353848&{}:{}&-0.290478697529&,\\C^{\prime\prime\prime}_{\mathbf{6}}&{}\approx{}&0.702908489980&{}:{}&0.102616799035&{}:{}&0.194474710985&. \end{alignedat} \]
6 (220)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6}}&{}\approx{}&0.605705924598&{}:{}&0.267684106880&{}:{}&0.126609968522&,\\ A^\prime_{\mathbf{6}}&{}\approx{}&0.667552905630&{}:{}&0.147754264164&{}:{}&0.184692830205&,\\B^\prime_{\mathbf{6}}&{}\approx{}&0.281999757518&{}:{}&0.248000646617&{}:{}&0.469999595864&,\\C^\prime_{\mathbf{6}}&{}\approx{}&1.476569900005&{}:{}&1.968759866674&{}:{}&-2.445329766679&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.473557987287&{}:{}&0.523937218551&{}:{}&0.002504794162&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.724030539754&{}:{}&0.274879020346&{}:{}&0.001090439899&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.558785949906&{}:{}&0.176034395957&{}:{}&0.265179654137&, \end{alignedat} \]
6 (220)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6}}&{}\approx{}&0.617207145668&{}:{}&0.424280042126&{}:{}&-0.041487187795&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.108380256645&{}:{}&-0.108380256645&,\\B^*_{\mathbf{6}}&{}\approx{}&1.072061402818&{}:{}&0.000000000000&{}:{}&-0.072061402818&,\\C^*_{\mathbf{6}}&{}\approx{}&0.592620968267&{}:{}&0.407379031733&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6}}&{}\approx{}&0.588856682171&{}:{}&0.381640981690&{}:{}&0.029502336139&,\\ A^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.473557987287&{}:{}&0.523937218551&{}:{}&0.002504794162&,\\B^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.724030539754&{}:{}&0.274879020346&{}:{}&0.001090439899&,\\C^{\prime\prime}_{\mathbf{6}}&{}\approx{}&0.558785949906&{}:{}&0.176034395957&{}:{}&0.265179654137&,\\ A^*_{\mathbf{6}}&{}\approx{}&0.000000000000&{}:{}&1.108380256645&{}:{}&-0.108380256645&,\\B^*_{\mathbf{6}}&{}\approx{}&1.072061402818&{}:{}&0.000000000000&{}:{}&-0.072061402818&,\\C^*_{\mathbf{6}}&{}\approx{}&0.592620968267&{}:{}&0.407379031733&{}:{}&0.000000000000&. \end{alignedat} \]
6 (220)