Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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5c(312)

Malfatti circles

5c (312)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&, \\ P_{\mathbf{5c}}&{}\approx{}&-0.221397430242&{}:{}&-0.137761006992&{}:{}&1.359158437234&, \\ P^-_{\mathbf{5c}}&{}\approx{}&-0.082404275863&{}:{}&0.083917200764&{}:{}&0.998487075099&, \\ P^+_{\mathbf{5c}}&{}\approx{}&-0.373517596898&{}:{}&-0.380375300182&{}:{}&1.753892897080&, \\ Q_{\mathbf{5c}}&{}\approx{}&-0.981883107052&{}:{}&0.577298464206&{}:{}&1.404584642846&, \\ I^\prime_{\mathbf{5c}}&{}\approx{}&-0.566126464975&{}:{}&-0.576520425929&{}:{}&2.142646890904&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{5c}}\)
\(P^-_{\mathbf{5c}}\)
\(P^+_{\mathbf{5c}}\)
\(Q_{\mathbf{5c}}\)
\(I^\prime_{\mathbf{5c}}\)
5c (312)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5c}}&{}\approx{}&-21.186745878783&{}:{}&-55.466864696957&{}:{}&77.653610575740&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.504632472192&{}:{}&0.327156703744&{}:{}&1.177475768448&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.198129682738&{}:{}&-0.330216137896&{}:{}&1.528345820634&, \\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.284541991663&{}:{}&-0.144883060921&{}:{}&1.429425052585&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.322343703908&{}:{}&-0.656523730898&{}:{}&1.978867434805&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.813554859361&{}:{}&-0.506221488417&{}:{}&2.319776347778&, \\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.135829464760&{}:{}&0.088059176551&{}:{}&1.047770288209&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.117661807738&{}:{}&-0.308038412655&{}:{}&1.425700220394&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.271476697554&{}:{}&0.276460951725&{}:{}&0.995015745829&, \\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.291287847478&{}:{}&0.708712152522&,\\B^*_{\mathbf{5c}}&{}\approx{}&-2.322875655532&{}:{}&0.000000000000&{}:{}&3.322875655532&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.426891688585&{}:{}&-1.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5c}}}{B^\prime_{\mathbf{5c}}}{C^\prime_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^*_{\mathbf{5c}}}{B^*_{\mathbf{5c}}}{C^*_{\mathbf{5c}}}\)
5c (312)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5c}}}}&{}\approx{}&-11.093372939391&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.168210824064&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.066043227579&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-21.186745878783&{}:{}&-55.466864696957&{}:{}&77.653610575740&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.504632472192&{}:{}&0.327156703744&{}:{}&1.177475768448&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.198129682738&{}:{}&-0.330216137896&{}:{}&1.528345820634&. \end{alignedat} \]
5c (312)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5c}}&{}\approx{}&-0.221397430242&{}:{}&-0.137761006992&{}:{}&1.359158437234&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.284541991663&{}:{}&-0.144883060921&{}:{}&1.429425052585&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.322343703908&{}:{}&-0.656523730898&{}:{}&1.978867434805&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.813554859361&{}:{}&-0.506221488417&{}:{}&2.319776347778&. \end{alignedat} \]
5c (312)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5c}}&{}\approx{}&-0.082404275863&{}:{}&0.083917200764&{}:{}&0.998487075099&,\\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.135829464760&{}:{}&0.088059176551&{}:{}&1.047770288209&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.117661807738&{}:{}&-0.308038412655&{}:{}&1.425700220394&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.271476697554&{}:{}&0.276460951725&{}:{}&0.995015745829&. \end{alignedat} \]
5c (312)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5c}}&{}\approx{}&-0.373517596898&{}:{}&-0.380375300182&{}:{}&1.753892897080&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-21.186745878783&{}:{}&-55.466864696957&{}:{}&77.653610575740&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.504632472192&{}:{}&0.327156703744&{}:{}&1.177475768448&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.198129682738&{}:{}&-0.330216137896&{}:{}&1.528345820634&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.284541991663&{}:{}&-0.144883060921&{}:{}&1.429425052585&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.322343703908&{}:{}&-0.656523730898&{}:{}&1.978867434805&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.813554859361&{}:{}&-0.506221488417&{}:{}&2.319776347778&, \end{alignedat} \]
5c (312)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5c}}&{}\approx{}&-0.981883107052&{}:{}&0.577298464206&{}:{}&1.404584642846&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.291287847478&{}:{}&0.708712152522&,\\B^*_{\mathbf{5c}}&{}\approx{}&-2.322875655532&{}:{}&0.000000000000&{}:{}&3.322875655532&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.426891688585&{}:{}&-1.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5c}}&{}\approx{}&-0.566126464975&{}:{}&-0.576520425929&{}:{}&2.142646890904&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.284541991663&{}:{}&-0.144883060921&{}:{}&1.429425052585&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.322343703908&{}:{}&-0.656523730898&{}:{}&1.978867434805&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.813554859361&{}:{}&-0.506221488417&{}:{}&2.319776347778&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.291287847478&{}:{}&0.708712152522&,\\B^*_{\mathbf{5c}}&{}\approx{}&-2.322875655532&{}:{}&0.000000000000&{}:{}&3.322875655532&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.426891688585&{}:{}&-1.426891688585&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)