Derousseau's Generalization of the Malfatti circles

Half of an Equilateral Triangle

\(A=30\degree\), \(B=60\degree\), \(C=90\degree\).


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\(\mathbf{2a}\) \((031)\)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.010860181204&{}:{}&0.032310928399&{}:{}&0.978549252805&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.014393487276&{}:{}&-0.010469765874&{}:{}&0.996076278598&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.032969699277&{}:{}&0.069765309886&{}:{}&0.963204389391&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.066879066048&{}:{}&-0.293292170627&{}:{}&1.226413104580&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.068053891494&{}:{}&0.154486118169&{}:{}&0.913567773324&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{2a}}\)
\(P^-_{\mathbf{2a}}\)
\(P^+_{\mathbf{2a}}\)
\(Q_{\mathbf{2a}}\)
\(I^\prime_{\mathbf{2a}}\)
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.129062533106&{}:{}&0.404203485070&{}:{}&0.466733981825&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-1.673032607476&{}:{}&-0.673032607476&{}:{}&3.346065214951&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.025638005274&{}:{}&0.044406327739&{}:{}&0.981231677535&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.050502102634&{}:{}&0.033578034680&{}:{}&1.016924067954&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.010324736120&{}:{}&0.080021399081&{}:{}&0.930303337039&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.091536296568&{}:{}&0.272336406608&{}:{}&0.819199889960&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.027122129900&{}:{}&-0.010910772286&{}:{}&1.038032902186&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.013636034257&{}:{}&0.042705907257&{}:{}&0.943658058486&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.257139226613&{}:{}&-0.187042059235&{}:{}&0.929902832622&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.314313139868&{}:{}&1.314313139868&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.051712263916&{}:{}&0.000000000000&{}:{}&0.948287736084&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.295385137588&{}:{}&1.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.637570475982&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&4.570810086343&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.070044333012&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.129062533106&{}:{}&0.404203485070&{}:{}&0.466733981825&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-1.673032607476&{}:{}&-0.673032607476&{}:{}&3.346065214951&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.025638005274&{}:{}&0.044406327739&{}:{}&0.981231677535&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.010860181204&{}:{}&0.032310928399&{}:{}&0.978549252805&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.050502102634&{}:{}&0.033578034680&{}:{}&1.016924067954&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.010324736120&{}:{}&0.080021399081&{}:{}&0.930303337039&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.091536296568&{}:{}&0.272336406608&{}:{}&0.819199889960&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.014393487276&{}:{}&-0.010469765874&{}:{}&0.996076278598&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.027122129900&{}:{}&-0.010910772286&{}:{}&1.038032902186&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.013636034257&{}:{}&0.042705907257&{}:{}&0.943658058486&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.257139226613&{}:{}&-0.187042059235&{}:{}&0.929902832622&. \end{alignedat} \]
2a (031)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.032969699277&{}:{}&0.069765309886&{}:{}&0.963204389391&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.129062533106&{}:{}&0.404203485070&{}:{}&0.466733981825&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-1.673032607476&{}:{}&-0.673032607476&{}:{}&3.346065214951&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.025638005274&{}:{}&0.044406327739&{}:{}&0.981231677535&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.050502102634&{}:{}&0.033578034680&{}:{}&1.016924067954&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.010324736120&{}:{}&0.080021399081&{}:{}&0.930303337039&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.091536296568&{}:{}&0.272336406608&{}:{}&0.819199889960&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.066879066048&{}:{}&-0.293292170627&{}:{}&1.226413104580&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.314313139868&{}:{}&1.314313139868&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.051712263916&{}:{}&0.000000000000&{}:{}&0.948287736084&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.295385137588&{}:{}&1.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.068053891494&{}:{}&0.154486118169&{}:{}&0.913567773324&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.050502102634&{}:{}&0.033578034680&{}:{}&1.016924067954&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.010324736120&{}:{}&0.080021399081&{}:{}&0.930303337039&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.091536296568&{}:{}&0.272336406608&{}:{}&0.819199889960&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.314313139868&{}:{}&1.314313139868&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.051712263916&{}:{}&0.000000000000&{}:{}&0.948287736084&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.295385137588&{}:{}&1.295385137588&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Hiroyasu Kamo