Derousseau's Generalization of the Malfatti circles

The Smallest Eisenstein Triangle

\(C=120\degree\).   \(a:b:c=3:5:7\).


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0c(110)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&, \\ P_{\mathbf{c}}&{}\approx{}&1.147698972970&{}:{}&1.534870451693&{}:{}&-1.682569424663&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.908822192989&{}:{}&1.087999788320&{}:{}&-0.996821981309&, \\ P^+_{\mathbf{c}}&{}\approx{}&1.337596543337&{}:{}&1.890114913920&{}:{}&-2.227711457257&, \\ Q_{\mathbf{c}}&{}\approx{}&0.859918296271&{}:{}&0.741212616596&{}:{}&-0.601130912867&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&1.554181635732&{}:{}&2.320320697670&{}:{}&-2.874502333401&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{c}}\)
\(P^-_{\mathbf{c}}\)
\(P^+_{\mathbf{c}}\)
\(Q_{\mathbf{c}}\)
\(I^\prime_{\mathbf{c}}\)
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.396259365476&{}:{}&0.990648413689&{}:{}&-1.386907779164&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.676196490926&{}:{}&1.901595321235&{}:{}&-1.577791812161&,\\C^\prime_{\mathbf{c}}&{}\approx{}&11.093372939391&{}:{}&18.488954898986&{}:{}&-28.582327838377&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.274694014690&{}:{}&2.854588071446&{}:{}&-3.129282086136&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.889412640990&{}:{}&1.880536635931&{}:{}&-2.769949276921&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.059408776929&{}:{}&1.416795924950&{}:{}&-1.476204701880&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.809277125324&{}:{}&2.275843805387&{}:{}&-2.085120930712&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.632191181593&{}:{}&1.158042441729&{}:{}&-1.790233623322&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.830667423392&{}:{}&0.994436522112&{}:{}&-0.825103945504&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&5.291287847478&{}:{}&-4.291287847478&,\\B^*_{\mathbf{c}}&{}\approx{}&3.322875655532&{}:{}&0.000000000000&{}:{}&-2.322875655532&,\\C^*_{\mathbf{c}}&{}\approx{}&0.537069323539&{}:{}&0.462930676461&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.198129682738&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.225398830309&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&3.697790979797&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&5.000000000000&{}:{}&-7.000000000000&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.396259365476&{}:{}&0.990648413689&{}:{}&-1.386907779164&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.676196490926&{}:{}&1.901595321235&{}:{}&-1.577791812161&,\\C^\prime_{\mathbf{c}}&{}\approx{}&11.093372939391&{}:{}&18.488954898986&{}:{}&-28.582327838377&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&1.147698972970&{}:{}&1.534870451693&{}:{}&-1.682569424663&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.274694014690&{}:{}&2.854588071446&{}:{}&-3.129282086136&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.889412640990&{}:{}&1.880536635931&{}:{}&-2.769949276921&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.059408776929&{}:{}&1.416795924950&{}:{}&-1.476204701880&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.908822192989&{}:{}&1.087999788320&{}:{}&-0.996821981309&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.809277125324&{}:{}&2.275843805387&{}:{}&-2.085120930712&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.632191181593&{}:{}&1.158042441729&{}:{}&-1.790233623322&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.830667423392&{}:{}&0.994436522112&{}:{}&-0.825103945504&. \end{alignedat} \]
0c (110)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&1.337596543337&{}:{}&1.890114913920&{}:{}&-2.227711457257&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.396259365476&{}:{}&0.990648413689&{}:{}&-1.386907779164&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.676196490926&{}:{}&1.901595321235&{}:{}&-1.577791812161&,\\C^\prime_{\mathbf{c}}&{}\approx{}&11.093372939391&{}:{}&18.488954898986&{}:{}&-28.582327838377&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.274694014690&{}:{}&2.854588071446&{}:{}&-3.129282086136&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.889412640990&{}:{}&1.880536635931&{}:{}&-2.769949276921&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.059408776929&{}:{}&1.416795924950&{}:{}&-1.476204701880&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.859918296271&{}:{}&0.741212616596&{}:{}&-0.601130912867&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&5.291287847478&{}:{}&-4.291287847478&,\\B^*_{\mathbf{c}}&{}\approx{}&3.322875655532&{}:{}&0.000000000000&{}:{}&-2.322875655532&,\\C^*_{\mathbf{c}}&{}\approx{}&0.537069323539&{}:{}&0.462930676461&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&1.554181635732&{}:{}&2.320320697670&{}:{}&-2.874502333401&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.274694014690&{}:{}&2.854588071446&{}:{}&-3.129282086136&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.889412640990&{}:{}&1.880536635931&{}:{}&-2.769949276921&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.059408776929&{}:{}&1.416795924950&{}:{}&-1.476204701880&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&5.291287847478&{}:{}&-4.291287847478&,\\B^*_{\mathbf{c}}&{}\approx{}&3.322875655532&{}:{}&0.000000000000&{}:{}&-2.322875655532&,\\C^*_{\mathbf{c}}&{}\approx{}&0.537069323539&{}:{}&0.462930676461&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)