Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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0b(101)

Malfatti circles

0b (101)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&, \\ P_{\mathbf{b}}&{}\approx{}&0.742299567528&{}:{}&-0.477775860227&{}:{}&0.735476292698&, \\ P^-_{\mathbf{b}}&{}\approx{}&0.740357468020&{}:{}&-0.346067511513&{}:{}&0.605710043493&, \\ P^+_{\mathbf{b}}&{}\approx{}&0.743590502517&{}:{}&-0.565323853981&{}:{}&0.821733351463&, \\ Q_{\mathbf{b}}&{}\approx{}&0.801819298444&{}:{}&-0.262617327447&{}:{}&0.460798029003&, \\ I^\prime_{\mathbf{b}}&{}\approx{}&0.736018478900&{}:{}&-0.681837030797&{}:{}&0.945818551897&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{b}}\)
\(P^-_{\mathbf{b}}\)
\(P^+_{\mathbf{b}}\)
\(Q_{\mathbf{b}}\)
\(I^\prime_{\mathbf{b}}\)
0b (101)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{b}}&{}\approx{}&0.921519886806&{}:{}&-0.313920452774&{}:{}&0.392400565968&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.194570142487&{}:{}&-2.185520379965&{}:{}&1.990950237478&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.348570869931&{}:{}&-0.464761159908&{}:{}&1.116190289977&, \\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.539668572619&{}:{}&-0.853453141685&{}:{}&1.313784569066&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.690539198596&{}:{}&-0.374730907662&{}:{}&0.684191709066&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.966475532745&{}:{}&-0.622065132791&{}:{}&0.655589600046&, \\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.421470565186&{}:{}&-0.771099558751&{}:{}&1.349628993565&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.676829591870&{}:{}&-0.230565454932&{}:{}&0.553735863062&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.043421047656&{}:{}&-0.487729429390&{}:{}&0.444308381734&, \\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.325140769936&{}:{}&2.325140769936&,\\B^*_{\mathbf{b}}&{}\approx{}&0.635045378369&{}:{}&0.000000000000&{}:{}&0.364954621631&,\\C^*_{\mathbf{b}}&{}\approx{}&1.487048159267&{}:{}&-0.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{b}}}{B^\prime_{\mathbf{b}}}{C^\prime_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{b}}}{B^{\prime\prime}_{\mathbf{b}}}{C^{\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{b}}}{B^{\prime\prime\prime}_{\mathbf{b}}}{C^{\prime\prime\prime}_{\mathbf{b}}}\)
\(\triangle{A^*_{\mathbf{b}}}{B^*_{\mathbf{b}}}{C^*_{\mathbf{b}}}\)
0b (101)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{b}}}}&{}\approx{}&0.313920452774&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{b}}}}&{}\approx{}&1.592760189982&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{b}}}}&{}\approx{}&0.464761159908&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.921519886806&{}:{}&-0.313920452774&{}:{}&0.392400565968&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.194570142487&{}:{}&-2.185520379965&{}:{}&1.990950237478&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.348570869931&{}:{}&-0.464761159908&{}:{}&1.116190289977&. \end{alignedat} \]
0b (101)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{b}}&{}\approx{}&0.742299567528&{}:{}&-0.477775860227&{}:{}&0.735476292698&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.539668572619&{}:{}&-0.853453141685&{}:{}&1.313784569066&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.690539198596&{}:{}&-0.374730907662&{}:{}&0.684191709066&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.966475532745&{}:{}&-0.622065132791&{}:{}&0.655589600046&. \end{alignedat} \]
0b (101)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{b}}&{}\approx{}&0.740357468020&{}:{}&-0.346067511513&{}:{}&0.605710043493&,\\ A^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.421470565186&{}:{}&-0.771099558751&{}:{}&1.349628993565&,\\B^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&0.676829591870&{}:{}&-0.230565454932&{}:{}&0.553735863062&,\\C^{\prime\prime\prime}_{\mathbf{b}}&{}\approx{}&1.043421047656&{}:{}&-0.487729429390&{}:{}&0.444308381734&. \end{alignedat} \]
0b (101)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{b}}&{}\approx{}&0.743590502517&{}:{}&-0.565323853981&{}:{}&0.821733351463&,\\ A^\prime_{\mathbf{b}}&{}\approx{}&0.921519886806&{}:{}&-0.313920452774&{}:{}&0.392400565968&,\\B^\prime_{\mathbf{b}}&{}\approx{}&1.194570142487&{}:{}&-2.185520379965&{}:{}&1.990950237478&,\\C^\prime_{\mathbf{b}}&{}\approx{}&0.348570869931&{}:{}&-0.464761159908&{}:{}&1.116190289977&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.539668572619&{}:{}&-0.853453141685&{}:{}&1.313784569066&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.690539198596&{}:{}&-0.374730907662&{}:{}&0.684191709066&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.966475532745&{}:{}&-0.622065132791&{}:{}&0.655589600046&, \end{alignedat} \]
0b (101)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{b}}&{}\approx{}&0.801819298444&{}:{}&-0.262617327447&{}:{}&0.460798029003&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.325140769936&{}:{}&2.325140769936&,\\B^*_{\mathbf{b}}&{}\approx{}&0.635045378369&{}:{}&0.000000000000&{}:{}&0.364954621631&,\\C^*_{\mathbf{b}}&{}\approx{}&1.487048159267&{}:{}&-0.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{b}}&{}\approx{}&0.736018478900&{}:{}&-0.681837030797&{}:{}&0.945818551897&,\\ A^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.539668572619&{}:{}&-0.853453141685&{}:{}&1.313784569066&,\\B^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.690539198596&{}:{}&-0.374730907662&{}:{}&0.684191709066&,\\C^{\prime\prime}_{\mathbf{b}}&{}\approx{}&0.966475532745&{}:{}&-0.622065132791&{}:{}&0.655589600046&,\\ A^*_{\mathbf{b}}&{}\approx{}&0.000000000000&{}:{}&-1.325140769936&{}:{}&2.325140769936&,\\B^*_{\mathbf{b}}&{}\approx{}&0.635045378369&{}:{}&0.000000000000&{}:{}&0.364954621631&,\\C^*_{\mathbf{b}}&{}\approx{}&1.487048159267&{}:{}&-0.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
0b (101)