Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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1a(013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.056366158180&{}:{}&0.933952934371&{}:{}&0.122413223809&, \\ P^-_{\mathbf{1a}}&{}\approx{}&0.002073696390&{}:{}&0.969162535490&{}:{}&0.028763768120&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.102619985859&{}:{}&0.906085328697&{}:{}&0.196534657163&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.118789171705&{}:{}&1.184420068139&{}:{}&-0.303209239844&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.178482548008&{}:{}&0.838915857229&{}:{}&0.339566690779&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{1a}}\)
\(P^-_{\mathbf{1a}}\)
\(P^+_{\mathbf{1a}}\)
\(Q_{\mathbf{1a}}\)
\(I^\prime_{\mathbf{1a}}\)
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&0.010385691552&{}:{}&0.439828581532&{}:{}&0.549785726915&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.065015054523&{}:{}&0.956656630318&{}:{}&0.108358424204&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.298631302823&{}:{}&1.731508403764&{}:{}&0.567122899059&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.123830617085&{}:{}&0.993599515126&{}:{}&0.130231101959&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.254770335235&{}:{}&0.701472770276&{}:{}&0.553297564960&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.052600895687&{}:{}&0.871564826561&{}:{}&0.181036069127&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.070665912037&{}:{}&1.039805530948&{}:{}&0.030860381089&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.017476331757&{}:{}&0.740113469424&{}:{}&0.242410198819&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.001918293208&{}:{}&0.896533319738&{}:{}&0.101548387054&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.344082517041&{}:{}&-0.344082517041&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.644122805635&{}:{}&0.000000000000&{}:{}&1.644122805635&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.091151265716&{}:{}&0.908848734284&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.659742872299&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.130030109045&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&2.597262605646&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&0.010385691552&{}:{}&0.439828581532&{}:{}&0.549785726915&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.065015054523&{}:{}&0.956656630318&{}:{}&0.108358424204&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.298631302823&{}:{}&1.731508403764&{}:{}&0.567122899059&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.056366158180&{}:{}&0.933952934371&{}:{}&0.122413223809&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.123830617085&{}:{}&0.993599515126&{}:{}&0.130231101959&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.254770335235&{}:{}&0.701472770276&{}:{}&0.553297564960&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.052600895687&{}:{}&0.871564826561&{}:{}&0.181036069127&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&0.002073696390&{}:{}&0.969162535490&{}:{}&0.028763768120&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.070665912037&{}:{}&1.039805530948&{}:{}&0.030860381089&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.017476331757&{}:{}&0.740113469424&{}:{}&0.242410198819&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&0.001918293208&{}:{}&0.896533319738&{}:{}&0.101548387054&. \end{alignedat} \]
1a (013)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.102619985859&{}:{}&0.906085328697&{}:{}&0.196534657163&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&0.010385691552&{}:{}&0.439828581532&{}:{}&0.549785726915&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.065015054523&{}:{}&0.956656630318&{}:{}&0.108358424204&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.298631302823&{}:{}&1.731508403764&{}:{}&0.567122899059&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.123830617085&{}:{}&0.993599515126&{}:{}&0.130231101959&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.254770335235&{}:{}&0.701472770276&{}:{}&0.553297564960&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.052600895687&{}:{}&0.871564826561&{}:{}&0.181036069127&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.118789171705&{}:{}&1.184420068139&{}:{}&-0.303209239844&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.344082517041&{}:{}&-0.344082517041&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.644122805635&{}:{}&0.000000000000&{}:{}&1.644122805635&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.091151265716&{}:{}&0.908848734284&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.178482548008&{}:{}&0.838915857229&{}:{}&0.339566690779&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.123830617085&{}:{}&0.993599515126&{}:{}&0.130231101959&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.254770335235&{}:{}&0.701472770276&{}:{}&0.553297564960&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.052600895687&{}:{}&0.871564826561&{}:{}&0.181036069127&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.344082517041&{}:{}&-0.344082517041&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.644122805635&{}:{}&0.000000000000&{}:{}&1.644122805635&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.091151265716&{}:{}&0.908848734284&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)