Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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6b(321)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.018293494781&{}:{}&1.023185050200&{}:{}&-0.004891555418&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.015683682597&{}:{}&0.933711277759&{}:{}&0.050605039644&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.055567499415&{}:{}&1.121340520015&{}:{}&-0.065773020600&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.425218285517&{}:{}&1.298270112994&{}:{}&0.126948172523&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.143510376500&{}:{}&1.239314196762&{}:{}&-0.095803820262&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{6b}}\)
\(P^-_{\mathbf{6b}}\)
\(P^+_{\mathbf{6b}}\)
\(Q_{\mathbf{6b}}\)
\(I^\prime_{\mathbf{6b}}\)
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&3.389140284973&{}:{}&9.556561139893&{}:{}&-11.945701424867&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.039240056597&{}:{}&1.104640150925&{}:{}&-0.065400094328&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.537910698151&{}:{}&0.717214264202&{}:{}&0.820696433950&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.073144307846&{}:{}&1.078299348982&{}:{}&-0.005155041136&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.263762503300&{}:{}&1.334290793143&{}:{}&-0.070528289843&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.020572626522&{}:{}&1.150660612000&{}:{}&-0.130087985478&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.034871709233&{}:{}&0.981667548186&{}:{}&0.053204161047&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.141917162483&{}:{}&0.400172293277&{}:{}&0.457910544240&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.017541018802&{}:{}&1.044285803246&{}:{}&-0.061826822047&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.910927207563&{}:{}&0.089072792437&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.425614793411&{}:{}&0.000000000000&{}:{}&-0.425614793411&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.487048159267&{}:{}&1.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-9.556561139893&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.052320075462&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.717214264202&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&3.389140284973&{}:{}&9.556561139893&{}:{}&-11.945701424867&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.039240056597&{}:{}&1.104640150925&{}:{}&-0.065400094328&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.537910698151&{}:{}&0.717214264202&{}:{}&0.820696433950&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.018293494781&{}:{}&1.023185050200&{}:{}&-0.004891555418&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.073144307846&{}:{}&1.078299348982&{}:{}&-0.005155041136&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.263762503300&{}:{}&1.334290793143&{}:{}&-0.070528289843&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.020572626522&{}:{}&1.150660612000&{}:{}&-0.130087985478&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.015683682597&{}:{}&0.933711277759&{}:{}&0.050605039644&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.034871709233&{}:{}&0.981667548186&{}:{}&0.053204161047&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.141917162483&{}:{}&0.400172293277&{}:{}&0.457910544240&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.017541018802&{}:{}&1.044285803246&{}:{}&-0.061826822047&. \end{alignedat} \]
6b (321)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.055567499415&{}:{}&1.121340520015&{}:{}&-0.065773020600&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&3.389140284973&{}:{}&9.556561139893&{}:{}&-11.945701424867&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.039240056597&{}:{}&1.104640150925&{}:{}&-0.065400094328&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.537910698151&{}:{}&0.717214264202&{}:{}&0.820696433950&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.073144307846&{}:{}&1.078299348982&{}:{}&-0.005155041136&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.263762503300&{}:{}&1.334290793143&{}:{}&-0.070528289843&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.020572626522&{}:{}&1.150660612000&{}:{}&-0.130087985478&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.425218285517&{}:{}&1.298270112994&{}:{}&0.126948172523&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.910927207563&{}:{}&0.089072792437&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.425614793411&{}:{}&0.000000000000&{}:{}&-0.425614793411&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.487048159267&{}:{}&1.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.143510376500&{}:{}&1.239314196762&{}:{}&-0.095803820262&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.073144307846&{}:{}&1.078299348982&{}:{}&-0.005155041136&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.263762503300&{}:{}&1.334290793143&{}:{}&-0.070528289843&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.020572626522&{}:{}&1.150660612000&{}:{}&-0.130087985478&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.910927207563&{}:{}&0.089072792437&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.425614793411&{}:{}&0.000000000000&{}:{}&-0.425614793411&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.487048159267&{}:{}&1.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)