Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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3a(033)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.011192571526&{}:{}&0.185454097941&{}:{}&0.825738473585&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.064906841236&{}:{}&0.110537082036&{}:{}&0.824556076729&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.069223141475&{}:{}&0.242583016208&{}:{}&0.826640125267&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.038687968720&{}:{}&0.385749019792&{}:{}&0.575563011488&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.067617961931&{}:{}&0.317822560974&{}:{}&0.749795400956&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{3a}}\)
\(P^-_{\mathbf{3a}}\)
\(P^+_{\mathbf{3a}}\)
\(Q_{\mathbf{3a}}\)
\(I^\prime_{\mathbf{3a}}\)
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.217492472739&{}:{}&0.347781123227&{}:{}&0.434726404034&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.479228616896&{}:{}&0.680514255403&{}:{}&0.798714361493&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.091524521636&{}:{}&0.122032695515&{}:{}&0.969491826121&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.153695764262&{}:{}&0.211589378012&{}:{}&0.942106386250&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.011258307065&{}:{}&0.180670155809&{}:{}&0.830588151256&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.028066991791&{}:{}&0.465052971294&{}:{}&0.563014020497&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.090804085441&{}:{}&0.128943624009&{}:{}&0.961860461432&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.065347799154&{}:{}&0.104494333547&{}:{}&0.830157867299&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.212683484620&{}:{}&0.362202370959&{}:{}&0.425114144421&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.401273475459&{}:{}&0.598726524541&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.062983975551&{}:{}&0.000000000000&{}:{}&0.937016024449&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.091151265716&{}:{}&0.908848734284&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.521671684841&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&0.958457233792&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.183049043273&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.217492472739&{}:{}&0.347781123227&{}:{}&0.434726404034&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.479228616896&{}:{}&0.680514255403&{}:{}&0.798714361493&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.091524521636&{}:{}&0.122032695515&{}:{}&0.969491826121&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.011192571526&{}:{}&0.185454097941&{}:{}&0.825738473585&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.153695764262&{}:{}&0.211589378012&{}:{}&0.942106386250&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.011258307065&{}:{}&0.180670155809&{}:{}&0.830588151256&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.028066991791&{}:{}&0.465052971294&{}:{}&0.563014020497&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.064906841236&{}:{}&0.110537082036&{}:{}&0.824556076729&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.090804085441&{}:{}&0.128943624009&{}:{}&0.961860461432&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.065347799154&{}:{}&0.104494333547&{}:{}&0.830157867299&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.212683484620&{}:{}&0.362202370959&{}:{}&0.425114144421&. \end{alignedat} \]
3a (033)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.069223141475&{}:{}&0.242583016208&{}:{}&0.826640125267&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.217492472739&{}:{}&0.347781123227&{}:{}&0.434726404034&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-0.479228616896&{}:{}&0.680514255403&{}:{}&0.798714361493&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.091524521636&{}:{}&0.122032695515&{}:{}&0.969491826121&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.153695764262&{}:{}&0.211589378012&{}:{}&0.942106386250&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.011258307065&{}:{}&0.180670155809&{}:{}&0.830588151256&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.028066991791&{}:{}&0.465052971294&{}:{}&0.563014020497&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.038687968720&{}:{}&0.385749019792&{}:{}&0.575563011488&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.401273475459&{}:{}&0.598726524541&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.062983975551&{}:{}&0.000000000000&{}:{}&0.937016024449&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.091151265716&{}:{}&0.908848734284&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.067617961931&{}:{}&0.317822560974&{}:{}&0.749795400956&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.153695764262&{}:{}&0.211589378012&{}:{}&0.942106386250&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.011258307065&{}:{}&0.180670155809&{}:{}&0.830588151256&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.028066991791&{}:{}&0.465052971294&{}:{}&0.563014020497&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.401273475459&{}:{}&0.598726524541&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.062983975551&{}:{}&0.000000000000&{}:{}&0.937016024449&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.091151265716&{}:{}&0.908848734284&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)