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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\(\mathbf{6a}\) \((231)\)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&, \\ P_{\mathbf{6a}}&{}\approx{}&1.001810031150&{}:{}&-0.001517493014&{}:{}&-0.000292538135&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.971101486441&{}:{}&0.011062424337&{}:{}&0.017836089222&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.033952390331&{}:{}&-0.014684780029&{}:{}&-0.019267610302&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.132541826361&{}:{}&-0.164925925633&{}:{}&0.032384099271&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.078539355889&{}:{}&-0.051410554602&{}:{}&-0.027128801287&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.032610683953&{}:{}&-0.013340734344&{}:{}&-0.019269949609&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&6.412331612121&{}:{}&8.481053547475&{}:{}&-13.893385159596&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.612568271036&{}:{}&-0.918852406554&{}:{}&1.306284135518&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.118305158587&{}:{}&-0.099184619976&{}:{}&-0.019120538611&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.026599403674&{}:{}&-0.026299626804&{}:{}&-0.000299776870&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.040061954065&{}:{}&-0.001575435163&{}:{}&-0.038486518902&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.224462493476&{}:{}&0.296877725876&{}:{}&0.478659780648&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.994582072662&{}:{}&-0.012849426625&{}:{}&0.018267353963&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.007323000053&{}:{}&0.011475046251&{}:{}&-0.018798046304&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.244331168209&{}:{}&-0.244331168209&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.972200722330&{}:{}&0.000000000000&{}:{}&0.027799277670&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.170445654633&{}:{}&-0.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.023716861057&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-17.099550965657&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-1.633515389429&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.032610683953&{}:{}&-0.013340734344&{}:{}&-0.019269949609&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&6.412331612121&{}:{}&8.481053547475&{}:{}&-13.893385159596&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.612568271036&{}:{}&-0.918852406554&{}:{}&1.306284135518&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.001810031150&{}:{}&-0.001517493014&{}:{}&-0.000292538135&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.118305158587&{}:{}&-0.099184619976&{}:{}&-0.019120538611&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.026599403674&{}:{}&-0.026299626804&{}:{}&-0.000299776870&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.040061954065&{}:{}&-0.001575435163&{}:{}&-0.038486518902&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.971101486441&{}:{}&0.011062424337&{}:{}&0.017836089222&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.224462493476&{}:{}&0.296877725876&{}:{}&0.478659780648&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.994582072662&{}:{}&-0.012849426625&{}:{}&0.018267353963&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.007323000053&{}:{}&0.011475046251&{}:{}&-0.018798046304&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.033952390331&{}:{}&-0.014684780029&{}:{}&-0.019267610302&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.032610683953&{}:{}&-0.013340734344&{}:{}&-0.019269949609&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&6.412331612121&{}:{}&8.481053547475&{}:{}&-13.893385159596&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.612568271036&{}:{}&-0.918852406554&{}:{}&1.306284135518&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.118305158587&{}:{}&-0.099184619976&{}:{}&-0.019120538611&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.026599403674&{}:{}&-0.026299626804&{}:{}&-0.000299776870&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.040061954065&{}:{}&-0.001575435163&{}:{}&-0.038486518902&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.132541826361&{}:{}&-0.164925925633&{}:{}&0.032384099271&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.244331168209&{}:{}&-0.244331168209&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.972200722330&{}:{}&0.000000000000&{}:{}&0.027799277670&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.170445654633&{}:{}&-0.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.078539355889&{}:{}&-0.051410554602&{}:{}&-0.027128801287&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.118305158587&{}:{}&-0.099184619976&{}:{}&-0.019120538611&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.026599403674&{}:{}&-0.026299626804&{}:{}&-0.000299776870&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.040061954065&{}:{}&-0.001575435163&{}:{}&-0.038486518902&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.244331168209&{}:{}&-0.244331168209&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.972200722330&{}:{}&0.000000000000&{}:{}&0.027799277670&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.170445654633&{}:{}&-0.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Hiroyasu Kamo