Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).

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7b (323)

Malfatti circles

7b (323)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&, \\ P_{\mathbf{7b}}&{}\approx{}&-0.008371417971&{}:{}&1.019056988049&{}:{}&-0.010685570077&, \\ P^-_{\mathbf{7b}}&{}\approx{}&0.008904333621&{}:{}&0.957091698573&{}:{}&0.034003967806&, \\ P^+_{\mathbf{7b}}&{}\approx{}&-0.026386686114&{}:{}&1.083674802186&{}:{}&-0.057288116072&, \\ Q_{\mathbf{7b}}&{}\approx{}&0.180701187852&{}:{}&0.727997852861&{}:{}&0.091300959287&, \\ I^\prime_{\mathbf{7b}}&{}\approx{}&-0.070073392212&{}:{}&1.197728780605&{}:{}&-0.127655388392&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{7b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{7b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{7b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{7b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{7b}}\) Radical center of the Malfatti circles
7b (323)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7b}}&{}\approx{}&1.623827207244&{}:{}&7.485926486933&{}:{}&-8.109753694177&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.019099492400&{}:{}&1.068758172640&{}:{}&-0.049658680240&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.295152490866&{}:{}&0.708365978077&{}:{}&0.586786512788&, \\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.035877340837&{}:{}&1.046854387310&{}:{}&-0.010977046473&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.129263863316&{}:{}&1.294260770012&{}:{}&-0.164996906696&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.009100540528&{}:{}&1.107813449509&{}:{}&-0.098712908981&, \\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.017560652705&{}:{}&0.982648685224&{}:{}&0.034911967481&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.106057335066&{}:{}&0.488929739669&{}:{}&0.405012925264&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.009662598159&{}:{}&1.038594562927&{}:{}&-0.048257161087&, \\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.888562075359&{}:{}&0.111437924641&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.664337358189&{}:{}&0.000000000000&{}:{}&0.335662641811&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.198857024995&{}:{}&0.801142975005&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7b}}}{B^\prime_{\mathbf{7b}}}{C^\prime_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7b}}}{B^{\prime\prime\prime}_{\mathbf{7b}}}{C^{\prime\prime\prime}_{\mathbf{7b}}}\)
\(\triangle{A^*_{\mathbf{7b}}}{B^*_{\mathbf{7b}}}{C^*_{\mathbf{7b}}}\)
7b (323)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7b}}}}&{}\approx{}&-3.742963243466&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.022919390880&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7b}}}}&{}\approx{}&-0.354182989039&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.833333333333&{}:{}&-2.000000000000&{}:{}&2.166666666667&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&1.623827207244&{}:{}&7.485926486933&{}:{}&-8.109753694177&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.019099492400&{}:{}&1.068758172640&{}:{}&-0.049658680240&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.295152490866&{}:{}&0.708365978077&{}:{}&0.586786512788&. \end{alignedat} \]
7b (323)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7b}}&{}\approx{}&-0.008371417971&{}:{}&1.019056988049&{}:{}&-0.010685570077&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.035877340837&{}:{}&1.046854387310&{}:{}&-0.010977046473&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.129263863316&{}:{}&1.294260770012&{}:{}&-0.164996906696&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.009100540528&{}:{}&1.107813449509&{}:{}&-0.098712908981&. \end{alignedat} \]
7b (323)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7b}}&{}\approx{}&0.008904333621&{}:{}&0.957091698573&{}:{}&0.034003967806&,\\ A^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.017560652705&{}:{}&0.982648685224&{}:{}&0.034911967481&,\\B^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.106057335066&{}:{}&0.488929739669&{}:{}&0.405012925264&,\\C^{\prime\prime\prime}_{\mathbf{7b}}&{}\approx{}&0.009662598159&{}:{}&1.038594562927&{}:{}&-0.048257161087&. \end{alignedat} \]
7b (323)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7b}}&{}\approx{}&-0.026386686114&{}:{}&1.083674802186&{}:{}&-0.057288116072&,\\ A^\prime_{\mathbf{7b}}&{}\approx{}&1.623827207244&{}:{}&7.485926486933&{}:{}&-8.109753694177&,\\B^\prime_{\mathbf{7b}}&{}\approx{}&-0.019099492400&{}:{}&1.068758172640&{}:{}&-0.049658680240&,\\C^\prime_{\mathbf{7b}}&{}\approx{}&-0.295152490866&{}:{}&0.708365978077&{}:{}&0.586786512788&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.035877340837&{}:{}&1.046854387310&{}:{}&-0.010977046473&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.129263863316&{}:{}&1.294260770012&{}:{}&-0.164996906696&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.009100540528&{}:{}&1.107813449509&{}:{}&-0.098712908981&, \end{alignedat} \]
7b (323)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7b}}&{}\approx{}&0.180701187852&{}:{}&0.727997852861&{}:{}&0.091300959287&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.888562075359&{}:{}&0.111437924641&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.664337358189&{}:{}&0.000000000000&{}:{}&0.335662641811&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.198857024995&{}:{}&0.801142975005&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7b}}&{}\approx{}&-0.070073392212&{}:{}&1.197728780605&{}:{}&-0.127655388392&,\\ A^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.035877340837&{}:{}&1.046854387310&{}:{}&-0.010977046473&,\\B^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.129263863316&{}:{}&1.294260770012&{}:{}&-0.164996906696&,\\C^{\prime\prime}_{\mathbf{7b}}&{}\approx{}&-0.009100540528&{}:{}&1.107813449509&{}:{}&-0.098712908981&,\\ A^*_{\mathbf{7b}}&{}\approx{}&0.000000000000&{}:{}&0.888562075359&{}:{}&0.111437924641&,\\B^*_{\mathbf{7b}}&{}\approx{}&0.664337358189&{}:{}&0.000000000000&{}:{}&0.335662641811&,\\C^*_{\mathbf{7b}}&{}\approx{}&0.198857024995&{}:{}&0.801142975005&{}:{}&0.000000000000&. \end{alignedat} \]
7b (323)