Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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4c(310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&, \\ P_{\mathbf{4c}}&{}\approx{}&0.210925647993&{}:{}&1.098299204994&{}:{}&-0.309224852988&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.090396537568&{}:{}&0.929688237120&{}:{}&-0.020084774688&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.321866223190&{}:{}&1.253496548562&{}:{}&-0.575362771752&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.714347000962&{}:{}&1.214978553222&{}:{}&0.499368447741&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.477011451243&{}:{}&1.405234547637&{}:{}&-0.882245998880&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{4c}}\)
\(P^-_{\mathbf{4c}}\)
\(P^+_{\mathbf{4c}}\)
\(Q_{\mathbf{4c}}\)
\(I^\prime_{\mathbf{4c}}\)
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&22.786745878783&{}:{}&54.466864696957&{}:{}&-76.253610575740&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.171299138859&{}:{}&1.228398851812&{}:{}&-0.399697990670&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.398129682738&{}:{}&0.663549471230&{}:{}&-0.061679153967&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.239535721483&{}:{}&1.058477329035&{}:{}&-0.298013050518&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.667598988612&{}:{}&1.311125984251&{}:{}&-0.978724972862&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.289224232884&{}:{}&1.506003409562&{}:{}&-0.795227642446&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.124748133085&{}:{}&0.894578130797&{}:{}&-0.019326263882&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.315646414989&{}:{}&0.754485553522&{}:{}&-0.070131968511&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.125973874293&{}:{}&1.295585342814&{}:{}&-0.421559217107&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.708712152522&{}:{}&0.291287847478&,\\B^*_{\mathbf{4c}}&{}\approx{}&3.322875655532&{}:{}&0.000000000000&{}:{}&-2.322875655532&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.426891688585&{}:{}&2.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&10.893372939391&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.057099712953&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.132709894246&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&22.786745878783&{}:{}&54.466864696957&{}:{}&-76.253610575740&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.171299138859&{}:{}&1.228398851812&{}:{}&-0.399697990670&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.398129682738&{}:{}&0.663549471230&{}:{}&-0.061679153967&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.210925647993&{}:{}&1.098299204994&{}:{}&-0.309224852988&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.239535721483&{}:{}&1.058477329035&{}:{}&-0.298013050518&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.667598988612&{}:{}&1.311125984251&{}:{}&-0.978724972862&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.289224232884&{}:{}&1.506003409562&{}:{}&-0.795227642446&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.090396537568&{}:{}&0.929688237120&{}:{}&-0.020084774688&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.124748133085&{}:{}&0.894578130797&{}:{}&-0.019326263882&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.315646414989&{}:{}&0.754485553522&{}:{}&-0.070131968511&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.125973874293&{}:{}&1.295585342814&{}:{}&-0.421559217107&. \end{alignedat} \]
4c (310)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.321866223190&{}:{}&1.253496548562&{}:{}&-0.575362771752&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&22.786745878783&{}:{}&54.466864696957&{}:{}&-76.253610575740&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.171299138859&{}:{}&1.228398851812&{}:{}&-0.399697990670&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.398129682738&{}:{}&0.663549471230&{}:{}&-0.061679153967&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.239535721483&{}:{}&1.058477329035&{}:{}&-0.298013050518&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.667598988612&{}:{}&1.311125984251&{}:{}&-0.978724972862&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.289224232884&{}:{}&1.506003409562&{}:{}&-0.795227642446&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.714347000962&{}:{}&1.214978553222&{}:{}&0.499368447741&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.708712152522&{}:{}&0.291287847478&,\\B^*_{\mathbf{4c}}&{}\approx{}&3.322875655532&{}:{}&0.000000000000&{}:{}&-2.322875655532&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.426891688585&{}:{}&2.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.477011451243&{}:{}&1.405234547637&{}:{}&-0.882245998880&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.239535721483&{}:{}&1.058477329035&{}:{}&-0.298013050518&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.667598988612&{}:{}&1.311125984251&{}:{}&-0.978724972862&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.289224232884&{}:{}&1.506003409562&{}:{}&-0.795227642446&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.708712152522&{}:{}&0.291287847478&,\\B^*_{\mathbf{4c}}&{}\approx{}&3.322875655532&{}:{}&0.000000000000&{}:{}&-2.322875655532&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.426891688585&{}:{}&2.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)