Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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4b(301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&, \\ P_{\mathbf{4b}}&{}\approx{}&0.029399088684&{}:{}&-0.018922515249&{}:{}&0.989523426565&, \\ P^-_{\mathbf{4b}}&{}\approx{}&-0.005737313314&{}:{}&0.028914688887&{}:{}&0.976822624427&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.061413458122&{}:{}&-0.062509161547&{}:{}&1.001095703425&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.373542825939&{}:{}&0.122345295038&{}:{}&1.251197530901&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.132215342733&{}:{}&-0.122482409476&{}:{}&0.990267066744&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{4b}}\)
\(P^-_{\mathbf{4b}}\)
\(P^+_{\mathbf{4b}}\)
\(Q_{\mathbf{4b}}\)
\(I^\prime_{\mathbf{4b}}\)
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&-0.889140284973&{}:{}&-7.556561139894&{}:{}&9.445701424867&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.289240056597&{}:{}&0.228693182409&{}:{}&0.482066760995&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.037910698151&{}:{}&-0.050547597535&{}:{}&1.012636899384&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.067035104274&{}:{}&-0.018188775902&{}:{}&0.951153671628&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.031750644233&{}:{}&-0.100423439295&{}:{}&1.068672795062&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.232020953184&{}:{}&-0.149338643517&{}:{}&0.917317690333&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.035085526767&{}:{}&0.027741042743&{}:{}&0.937173430490&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.006220486844&{}:{}&-0.052866223648&{}:{}&1.059086710492&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.068198073694&{}:{}&0.343702004007&{}:{}&0.724496069687&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.089072792437&{}:{}&0.910927207563&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.425614793411&{}:{}&0.000000000000&{}:{}&1.425614793411&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.487048159267&{}:{}&-0.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&7.556561139894&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.385653408796&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.050547597535&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-0.889140284973&{}:{}&-7.556561139894&{}:{}&9.445701424867&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.289240056597&{}:{}&0.228693182409&{}:{}&0.482066760995&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.037910698151&{}:{}&-0.050547597535&{}:{}&1.012636899384&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.029399088684&{}:{}&-0.018922515249&{}:{}&0.989523426565&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.067035104274&{}:{}&-0.018188775902&{}:{}&0.951153671628&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.031750644233&{}:{}&-0.100423439295&{}:{}&1.068672795062&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.232020953184&{}:{}&-0.149338643517&{}:{}&0.917317690333&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&-0.005737313314&{}:{}&0.028914688887&{}:{}&0.976822624427&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.035085526767&{}:{}&0.027741042743&{}:{}&0.937173430490&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.006220486844&{}:{}&-0.052866223648&{}:{}&1.059086710492&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.068198073694&{}:{}&0.343702004007&{}:{}&0.724496069687&. \end{alignedat} \]
4b (301)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.061413458122&{}:{}&-0.062509161547&{}:{}&1.001095703425&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-0.889140284973&{}:{}&-7.556561139894&{}:{}&9.445701424867&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.289240056597&{}:{}&0.228693182409&{}:{}&0.482066760995&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.037910698151&{}:{}&-0.050547597535&{}:{}&1.012636899384&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.067035104274&{}:{}&-0.018188775902&{}:{}&0.951153671628&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.031750644233&{}:{}&-0.100423439295&{}:{}&1.068672795062&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.232020953184&{}:{}&-0.149338643517&{}:{}&0.917317690333&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.373542825939&{}:{}&0.122345295038&{}:{}&1.251197530901&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.089072792437&{}:{}&0.910927207563&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.425614793411&{}:{}&0.000000000000&{}:{}&1.425614793411&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.487048159267&{}:{}&-0.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.132215342733&{}:{}&-0.122482409476&{}:{}&0.990267066744&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.067035104274&{}:{}&-0.018188775902&{}:{}&0.951153671628&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.031750644233&{}:{}&-0.100423439295&{}:{}&1.068672795062&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.232020953184&{}:{}&-0.149338643517&{}:{}&0.917317690333&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.089072792437&{}:{}&0.910927207563&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.425614793411&{}:{}&0.000000000000&{}:{}&1.425614793411&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.487048159267&{}:{}&-0.487048159267&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)