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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\(\mathbf{6b}\) \((321)\)

Malfatti circles

6b (321)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&, \\ P_{\mathbf{6b}}&{}\approx{}&-0.002950900545&{}:{}&1.004113042733&{}:{}&-0.001162142188&, \\ P^-_{\mathbf{6b}}&{}\approx{}&0.008816918941&{}:{}&0.966950383971&{}:{}&0.024232697089&, \\ P^+_{\mathbf{6b}}&{}\approx{}&-0.015196724615&{}:{}&1.042785235292&{}:{}&-0.027588510678&, \\ Q_{\mathbf{6b}}&{}\approx{}&-0.240853036775&{}:{}&1.187413894337&{}:{}&0.053439142439&, \\ I^\prime_{\mathbf{6b}}&{}\approx{}&-0.048382754863&{}:{}&1.093075701734&{}:{}&-0.044692946871&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{6b}}\)
\(P^-_{\mathbf{6b}}\)
\(P^+_{\mathbf{6b}}\)
\(Q_{\mathbf{6b}}\)
\(I^\prime_{\mathbf{6b}}\)
6b (321)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6b}}&{}\approx{}&8.728748213577&{}:{}&17.389683480548&{}:{}&-25.118431694126&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.012593428996&{}:{}&1.039879191820&{}:{}&-0.027285762824&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.389201689055&{}:{}&0.583802533582&{}:{}&0.805399155473&, \\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.024397428520&{}:{}&1.025584421291&{}:{}&-0.001186992771&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.093038649228&{}:{}&1.129679714459&{}:{}&-0.036641065231&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.003108148839&{}:{}&1.057620458610&{}:{}&-0.054512309771&, \\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.011955641829&{}:{}&0.987215091870&{}:{}&0.024740549959&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.174195922355&{}:{}&0.347038530524&{}:{}&0.478765547120&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.009277108564&{}:{}&1.017419321678&{}:{}&-0.026696430241&, \\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.956933544219&{}:{}&0.043066455781&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.285139704438&{}:{}&0.000000000000&{}:{}&-0.285139704438&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.254450661943&{}:{}&1.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6b}}}{B^\prime_{\mathbf{6b}}}{C^\prime_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6b}}}{B^{\prime\prime\prime}_{\mathbf{6b}}}{C^{\prime\prime\prime}_{\mathbf{6b}}}\)
\(\triangle{A^*_{\mathbf{6b}}}{B^*_{\mathbf{6b}}}{C^*_{\mathbf{6b}}}\)
6b (321)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6b}}}}&{}\approx{}&-19.321870533943&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.020989048326&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6b}}}}&{}\approx{}&-0.648669481758&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&8.728748213577&{}:{}&17.389683480548&{}:{}&-25.118431694126&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.012593428996&{}:{}&1.039879191820&{}:{}&-0.027285762824&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.389201689055&{}:{}&0.583802533582&{}:{}&0.805399155473&. \end{alignedat} \]
6b (321)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6b}}&{}\approx{}&-0.002950900545&{}:{}&1.004113042733&{}:{}&-0.001162142188&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.024397428520&{}:{}&1.025584421291&{}:{}&-0.001186992771&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.093038649228&{}:{}&1.129679714459&{}:{}&-0.036641065231&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.003108148839&{}:{}&1.057620458610&{}:{}&-0.054512309771&. \end{alignedat} \]
6b (321)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6b}}&{}\approx{}&0.008816918941&{}:{}&0.966950383971&{}:{}&0.024232697089&,\\ A^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.011955641829&{}:{}&0.987215091870&{}:{}&0.024740549959&,\\B^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.174195922355&{}:{}&0.347038530524&{}:{}&0.478765547120&,\\C^{\prime\prime\prime}_{\mathbf{6b}}&{}\approx{}&0.009277108564&{}:{}&1.017419321678&{}:{}&-0.026696430241&. \end{alignedat} \]
6b (321)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6b}}&{}\approx{}&-0.015196724615&{}:{}&1.042785235292&{}:{}&-0.027588510678&,\\ A^\prime_{\mathbf{6b}}&{}\approx{}&8.728748213577&{}:{}&17.389683480548&{}:{}&-25.118431694126&,\\B^\prime_{\mathbf{6b}}&{}\approx{}&-0.012593428996&{}:{}&1.039879191820&{}:{}&-0.027285762824&,\\C^\prime_{\mathbf{6b}}&{}\approx{}&-0.389201689055&{}:{}&0.583802533582&{}:{}&0.805399155473&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.024397428520&{}:{}&1.025584421291&{}:{}&-0.001186992771&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.093038649228&{}:{}&1.129679714459&{}:{}&-0.036641065231&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.003108148839&{}:{}&1.057620458610&{}:{}&-0.054512309771&, \end{alignedat} \]
6b (321)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6b}}&{}\approx{}&-0.240853036775&{}:{}&1.187413894337&{}:{}&0.053439142439&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.956933544219&{}:{}&0.043066455781&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.285139704438&{}:{}&0.000000000000&{}:{}&-0.285139704438&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.254450661943&{}:{}&1.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6b}}&{}\approx{}&-0.048382754863&{}:{}&1.093075701734&{}:{}&-0.044692946871&,\\ A^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.024397428520&{}:{}&1.025584421291&{}:{}&-0.001186992771&,\\B^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.093038649228&{}:{}&1.129679714459&{}:{}&-0.036641065231&,\\C^{\prime\prime}_{\mathbf{6b}}&{}\approx{}&-0.003108148839&{}:{}&1.057620458610&{}:{}&-0.054512309771&,\\ A^*_{\mathbf{6b}}&{}\approx{}&0.000000000000&{}:{}&0.956933544219&{}:{}&0.043066455781&,\\B^*_{\mathbf{6b}}&{}\approx{}&1.285139704438&{}:{}&0.000000000000&{}:{}&-0.285139704438&,\\C^*_{\mathbf{6b}}&{}\approx{}&-0.254450661943&{}:{}&1.254450661943&{}:{}&0.000000000000&. \end{alignedat} \]
6b (321)

Hiroyasu Kamo