Derousseau's Generalization of the Malfatti circles

Test Case in ETC

The test case used in “Encyclopedis of Triangle Centers&rdquot; to search for triangle centers.

\(a:b:c=6:9:13\).


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\(\mathbf{2a}\) \((031)\)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.002277009832&{}:{}&0.007669508093&{}:{}&0.994607501739&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.010303424101&{}:{}&-0.011057558742&{}:{}&1.000754134641&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.014061898459&{}:{}&0.025212336752&{}:{}&0.988849561707&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.034358139274&{}:{}&-0.235936403903&{}:{}&1.201578264628&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.031238765872&{}:{}&0.070216762222&{}:{}&0.961022003651&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{2a}}\)
\(P^-_{\mathbf{2a}}\)
\(P^+_{\mathbf{2a}}\)
\(Q_{\mathbf{2a}}\)
\(I^\prime_{\mathbf{2a}}\)
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.181675030333&{}:{}&0.334769305773&{}:{}&0.483555663894&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-3.412331612121&{}:{}&-2.981053547475&{}:{}&7.393385159596&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.012568271036&{}:{}&0.018852406554&{}:{}&0.993715864482&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.025044299121&{}:{}&0.007843725308&{}:{}&1.017200573812&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.002212703037&{}:{}&0.035694680695&{}:{}&0.966518022342&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.038956596460&{}:{}&0.131215038058&{}:{}&0.907741558401&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.012954739385&{}:{}&-0.011317414656&{}:{}&1.024272154041&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.010002901988&{}:{}&0.018432167305&{}:{}&0.971564930708&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.386330225039&{}:{}&-0.414606747734&{}:{}&1.028276522696&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.244331168209&{}:{}&1.244331168209&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.027799277670&{}:{}&0.000000000000&{}:{}&0.972200722330&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.170445654633&{}:{}&1.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.595145432485&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&9.099550965657&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.033515389429&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.181675030333&{}:{}&0.334769305773&{}:{}&0.483555663894&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-3.412331612121&{}:{}&-2.981053547475&{}:{}&7.393385159596&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.012568271036&{}:{}&0.018852406554&{}:{}&0.993715864482&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.002277009832&{}:{}&0.007669508093&{}:{}&0.994607501739&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.025044299121&{}:{}&0.007843725308&{}:{}&1.017200573812&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.002212703037&{}:{}&0.035694680695&{}:{}&0.966518022342&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.038956596460&{}:{}&0.131215038058&{}:{}&0.907741558401&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.010303424101&{}:{}&-0.011057558742&{}:{}&1.000754134641&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.012954739385&{}:{}&-0.011317414656&{}:{}&1.024272154041&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.010002901988&{}:{}&0.018432167305&{}:{}&0.971564930708&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.386330225039&{}:{}&-0.414606747734&{}:{}&1.028276522696&. \end{alignedat} \]
2a (031)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.014061898459&{}:{}&0.025212336752&{}:{}&0.988849561707&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.181675030333&{}:{}&0.334769305773&{}:{}&0.483555663894&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-3.412331612121&{}:{}&-2.981053547475&{}:{}&7.393385159596&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.012568271036&{}:{}&0.018852406554&{}:{}&0.993715864482&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.025044299121&{}:{}&0.007843725308&{}:{}&1.017200573812&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.002212703037&{}:{}&0.035694680695&{}:{}&0.966518022342&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.038956596460&{}:{}&0.131215038058&{}:{}&0.907741558401&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.034358139274&{}:{}&-0.235936403903&{}:{}&1.201578264628&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.244331168209&{}:{}&1.244331168209&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.027799277670&{}:{}&0.000000000000&{}:{}&0.972200722330&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.170445654633&{}:{}&1.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.031238765872&{}:{}&0.070216762222&{}:{}&0.961022003651&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.025044299121&{}:{}&0.007843725308&{}:{}&1.017200573812&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.002212703037&{}:{}&0.035694680695&{}:{}&0.966518022342&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.038956596460&{}:{}&0.131215038058&{}:{}&0.907741558401&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.244331168209&{}:{}&1.244331168209&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.027799277670&{}:{}&0.000000000000&{}:{}&0.972200722330&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.170445654633&{}:{}&1.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Hiroyasu Kamo