Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 120° Top Angle

\(A=30\degree\), \(B=30\degree\), \(C=120\degree\).


[Top] > Isosceles Triangle with 120° Top Angle > 6a (231)

6a(231)

Malfatti circles

6a (231)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{6a}}&{}\approx{}&1.002698292943&{}:{}&-0.002119202410&{}:{}&-0.000579090533&, \\ P^-_{\mathbf{6a}}&{}\approx{}&0.969603343034&{}:{}&0.010018091086&{}:{}&0.020378565880&, \\ P^+_{\mathbf{6a}}&{}\approx{}&1.037240241488&{}:{}&-0.014787170440&{}:{}&-0.022453071048&, \\ Q_{\mathbf{6a}}&{}\approx{}&1.157804993949&{}:{}&-0.198647931795&{}:{}&0.040842937846&, \\ I^\prime_{\mathbf{6a}}&{}\approx{}&1.084129032472&{}:{}&-0.049840461676&{}:{}&-0.034288570796&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{6a}}\)
\(P^-_{\mathbf{6a}}\)
\(P^+_{\mathbf{6a}}\)
\(Q_{\mathbf{6a}}\)
\(I^\prime_{\mathbf{6a}}\)
6a (231)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6a}}&{}\approx{}&1.035276180410&{}:{}&-0.012911978179&{}:{}&-0.022364202232&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&10.375993078883&{}:{}&8.595754112725&{}:{}&-17.971747191608&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.593412988719&{}:{}&-0.593412988719&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.122622141240&{}:{}&-0.096305754298&{}:{}&-0.026316386942&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.025866493855&{}:{}&-0.025274022940&{}:{}&-0.000592470915&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.046885405069&{}:{}&-0.002212591852&{}:{}&-0.044672813216&, \\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.284608852862&{}:{}&0.235777693654&{}:{}&0.479613453484&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.991526993178&{}:{}&-0.012366337738&{}:{}&0.020839344560&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.011398467292&{}:{}&0.010449924747&{}:{}&-0.021848392040&, \\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.258819045103&{}:{}&-0.258819045103&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.965925826289&{}:{}&0.000000000000&{}:{}&0.034074173711&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6a}}}{B^\prime_{\mathbf{6a}}}{C^\prime_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6a}}}{B^{\prime\prime\prime}_{\mathbf{6a}}}{C^{\prime\prime\prime}_{\mathbf{6a}}}\)
\(\triangle{A^*_{\mathbf{6a}}}{B^*_{\mathbf{6a}}}{C^*_{\mathbf{6a}}}\)
6a (231)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6a}}}}&{}\approx{}&-0.022364202232&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6a}}}}&{}\approx{}&-17.971747191608&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6a}}}}&{}\approx{}&-1.027821446333&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.035276180410&{}:{}&-0.012911978179&{}:{}&-0.022364202232&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&10.375993078883&{}:{}&8.595754112725&{}:{}&-17.971747191608&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.593412988719&{}:{}&-0.593412988719&{}:{}&1.000000000000&. \end{alignedat} \]
6a (231)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6a}}&{}\approx{}&1.002698292943&{}:{}&-0.002119202410&{}:{}&-0.000579090533&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.122622141240&{}:{}&-0.096305754298&{}:{}&-0.026316386942&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.025866493855&{}:{}&-0.025274022940&{}:{}&-0.000592470915&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.046885405069&{}:{}&-0.002212591852&{}:{}&-0.044672813216&. \end{alignedat} \]
6a (231)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6a}}&{}\approx{}&0.969603343034&{}:{}&0.010018091086&{}:{}&0.020378565880&,\\ A^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.284608852862&{}:{}&0.235777693654&{}:{}&0.479613453484&,\\B^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&0.991526993178&{}:{}&-0.012366337738&{}:{}&0.020839344560&,\\C^{\prime\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.011398467292&{}:{}&0.010449924747&{}:{}&-0.021848392040&. \end{alignedat} \]
6a (231)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6a}}&{}\approx{}&1.037240241488&{}:{}&-0.014787170440&{}:{}&-0.022453071048&,\\ A^\prime_{\mathbf{6a}}&{}\approx{}&1.035276180410&{}:{}&-0.012911978179&{}:{}&-0.022364202232&,\\B^\prime_{\mathbf{6a}}&{}\approx{}&10.375993078883&{}:{}&8.595754112725&{}:{}&-17.971747191608&,\\C^\prime_{\mathbf{6a}}&{}\approx{}&0.593412988719&{}:{}&-0.593412988719&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.122622141240&{}:{}&-0.096305754298&{}:{}&-0.026316386942&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.025866493855&{}:{}&-0.025274022940&{}:{}&-0.000592470915&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.046885405069&{}:{}&-0.002212591852&{}:{}&-0.044672813216&, \end{alignedat} \]
6a (231)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6a}}&{}\approx{}&1.157804993949&{}:{}&-0.198647931795&{}:{}&0.040842937846&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.258819045103&{}:{}&-0.258819045103&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.965925826289&{}:{}&0.000000000000&{}:{}&0.034074173711&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6a}}&{}\approx{}&1.084129032472&{}:{}&-0.049840461676&{}:{}&-0.034288570796&,\\ A^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.122622141240&{}:{}&-0.096305754298&{}:{}&-0.026316386942&,\\B^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.025866493855&{}:{}&-0.025274022940&{}:{}&-0.000592470915&,\\C^{\prime\prime}_{\mathbf{6a}}&{}\approx{}&1.046885405069&{}:{}&-0.002212591852&{}:{}&-0.044672813216&,\\ A^*_{\mathbf{6a}}&{}\approx{}&0.000000000000&{}:{}&1.258819045103&{}:{}&-0.258819045103&,\\B^*_{\mathbf{6a}}&{}\approx{}&0.965925826289&{}:{}&0.000000000000&{}:{}&0.034074173711&,\\C^*_{\mathbf{6a}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
6a (231)