Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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5b(303)

Malfatti circles

5b (303)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&, \\ P_{\mathbf{5b}}&{}\approx{}&0.100910273272&{}:{}&-0.017528356780&{}:{}&0.916618083508&, \\ P^-_{\mathbf{5b}}&{}\approx{}&0.042234659609&{}:{}&0.071283904759&{}:{}&0.886481435632&, \\ P^+_{\mathbf{5b}}&{}\approx{}&0.150601952502&{}:{}&-0.092742404604&{}:{}&0.942140452101&, \\ Q_{\mathbf{5b}}&{}\approx{}&0.343617241206&{}:{}&0.058465845233&{}:{}&0.597916913561&, \\ I^\prime_{\mathbf{5b}}&{}\approx{}&0.226775075125&{}:{}&-0.109135891114&{}:{}&0.882360815989&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{5b}}\)
\(P^-_{\mathbf{5b}}\)
\(P^+_{\mathbf{5b}}\)
\(Q_{\mathbf{5b}}\)
\(I^\prime_{\mathbf{5b}}\)
5b (303)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5b}}&{}\approx{}&0.651429130069&{}:{}&-1.394283479725&{}:{}&1.742854349656&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.268955349076&{}:{}&0.282785735798&{}:{}&0.448258915126&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.078480113194&{}:{}&-0.104640150925&{}:{}&1.026160037731&, \\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.121505422052&{}:{}&-0.017126840552&{}:{}&0.895621418500&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.118477795779&{}:{}&-0.194670403171&{}:{}&1.076192607392&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.347202003362&{}:{}&-0.060309821709&{}:{}&0.713107818347&, \\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.066107671366&{}:{}&0.069507100541&{}:{}&0.864385228093&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.050380157362&{}:{}&-0.107830948716&{}:{}&1.057450791353&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.186454531542&{}:{}&0.314699045554&{}:{}&0.498846422904&, \\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.089072792437&{}:{}&0.910927207563&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.364954621631&{}:{}&0.000000000000&{}:{}&0.635045378369&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.854592627233&{}:{}&0.145407372767&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5b}}}{B^\prime_{\mathbf{5b}}}{C^\prime_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^*_{\mathbf{5b}}}{B^*_{\mathbf{5b}}}{C^*_{\mathbf{5b}}}\)
5b (303)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5b}}}}&{}\approx{}&1.394283479725&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5b}}}}&{}\approx{}&0.358607132101&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5b}}}}&{}\approx{}&0.104640150925&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.750000000000&{}:{}&-1.000000000000&{}:{}&1.250000000000&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.651429130069&{}:{}&-1.394283479725&{}:{}&1.742854349656&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.268955349076&{}:{}&0.282785735798&{}:{}&0.448258915126&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.078480113194&{}:{}&-0.104640150925&{}:{}&1.026160037731&. \end{alignedat} \]
5b (303)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5b}}&{}\approx{}&0.100910273272&{}:{}&-0.017528356780&{}:{}&0.916618083508&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.121505422052&{}:{}&-0.017126840552&{}:{}&0.895621418500&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.118477795779&{}:{}&-0.194670403171&{}:{}&1.076192607392&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.347202003362&{}:{}&-0.060309821709&{}:{}&0.713107818347&. \end{alignedat} \]
5b (303)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5b}}&{}\approx{}&0.042234659609&{}:{}&0.071283904759&{}:{}&0.886481435632&,\\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.066107671366&{}:{}&0.069507100541&{}:{}&0.864385228093&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.050380157362&{}:{}&-0.107830948716&{}:{}&1.057450791353&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.186454531542&{}:{}&0.314699045554&{}:{}&0.498846422904&. \end{alignedat} \]
5b (303)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5b}}&{}\approx{}&0.150601952502&{}:{}&-0.092742404604&{}:{}&0.942140452101&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.651429130069&{}:{}&-1.394283479725&{}:{}&1.742854349656&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.268955349076&{}:{}&0.282785735798&{}:{}&0.448258915126&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.078480113194&{}:{}&-0.104640150925&{}:{}&1.026160037731&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.121505422052&{}:{}&-0.017126840552&{}:{}&0.895621418500&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.118477795779&{}:{}&-0.194670403171&{}:{}&1.076192607392&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.347202003362&{}:{}&-0.060309821709&{}:{}&0.713107818347&, \end{alignedat} \]
5b (303)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5b}}&{}\approx{}&0.343617241206&{}:{}&0.058465845233&{}:{}&0.597916913561&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.089072792437&{}:{}&0.910927207563&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.364954621631&{}:{}&0.000000000000&{}:{}&0.635045378369&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.854592627233&{}:{}&0.145407372767&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5b}}&{}\approx{}&0.226775075125&{}:{}&-0.109135891114&{}:{}&0.882360815989&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.121505422052&{}:{}&-0.017126840552&{}:{}&0.895621418500&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.118477795779&{}:{}&-0.194670403171&{}:{}&1.076192607392&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.347202003362&{}:{}&-0.060309821709&{}:{}&0.713107818347&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.089072792437&{}:{}&0.910927207563&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.364954621631&{}:{}&0.000000000000&{}:{}&0.635045378369&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.854592627233&{}:{}&0.145407372767&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)