Derousseau's Generalization of the Malfatti circles

Test Case in ETC

The test case used in “Encyclopedis of Triangle Centers&rdquot; to search for triangle centers.

\(a:b:c=6:9:13\).


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\(\mathbf{1b}\) \((103)\)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&, \\ P_{\mathbf{1b}}&{}\approx{}&0.873433919452&{}:{}&-0.217415151806&{}:{}&0.343981232354&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.925266762137&{}:{}&-0.088022598289&{}:{}&0.162755836152&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.835850063223&{}:{}&-0.311237347861&{}:{}&0.475387284638&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.097132208512&{}:{}&0.146293430416&{}:{}&-0.243425638928&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.761795137361&{}:{}&-0.465493587296&{}:{}&0.703698449936&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.924257658374&{}:{}&-0.170420268659&{}:{}&0.246162610286&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.417927945364&{}:{}&-0.323438493651&{}:{}&0.905510548288&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.074140290046&{}:{}&-1.611210435069&{}:{}&1.537070145023&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.601732861538&{}:{}&-0.684143097369&{}:{}&1.082410235832&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.938594545532&{}:{}&-0.308237744934&{}:{}&0.369643199402&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.816032991599&{}:{}&-0.203126914121&{}:{}&0.387093922522&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.603475761431&{}:{}&-0.467035749577&{}:{}&0.863559988147&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.008557181205&{}:{}&-0.185963929238&{}:{}&0.177406748033&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.853822692194&{}:{}&-0.081225971710&{}:{}&0.227403279516&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.506126882702&{}:{}&2.506126882702&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.285139704438&{}:{}&0.000000000000&{}:{}&-0.285139704438&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.882346458175&{}:{}&0.117653541825&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.189355854066&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.696546575606&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&1.790233816743&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.600000000000&{}:{}&-0.900000000000&{}:{}&1.300000000000&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.924257658374&{}:{}&-0.170420268659&{}:{}&0.246162610286&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.417927945364&{}:{}&-0.323438493651&{}:{}&0.905510548288&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.074140290046&{}:{}&-1.611210435069&{}:{}&1.537070145023&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.873433919452&{}:{}&-0.217415151806&{}:{}&0.343981232354&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.601732861538&{}:{}&-0.684143097369&{}:{}&1.082410235832&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.938594545532&{}:{}&-0.308237744934&{}:{}&0.369643199402&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.816032991599&{}:{}&-0.203126914121&{}:{}&0.387093922522&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.925266762137&{}:{}&-0.088022598289&{}:{}&0.162755836152&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.603475761431&{}:{}&-0.467035749577&{}:{}&0.863559988147&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.008557181205&{}:{}&-0.185963929238&{}:{}&0.177406748033&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.853822692194&{}:{}&-0.081225971710&{}:{}&0.227403279516&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.835850063223&{}:{}&-0.311237347861&{}:{}&0.475387284638&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.924257658374&{}:{}&-0.170420268659&{}:{}&0.246162610286&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.417927945364&{}:{}&-0.323438493651&{}:{}&0.905510548288&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.074140290046&{}:{}&-1.611210435069&{}:{}&1.537070145023&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.601732861538&{}:{}&-0.684143097369&{}:{}&1.082410235832&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.938594545532&{}:{}&-0.308237744934&{}:{}&0.369643199402&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.816032991599&{}:{}&-0.203126914121&{}:{}&0.387093922522&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.097132208512&{}:{}&0.146293430416&{}:{}&-0.243425638928&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.506126882702&{}:{}&2.506126882702&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.285139704438&{}:{}&0.000000000000&{}:{}&-0.285139704438&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.882346458175&{}:{}&0.117653541825&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.761795137361&{}:{}&-0.465493587296&{}:{}&0.703698449936&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.601732861538&{}:{}&-0.684143097369&{}:{}&1.082410235832&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.938594545532&{}:{}&-0.308237744934&{}:{}&0.369643199402&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.816032991599&{}:{}&-0.203126914121&{}:{}&0.387093922522&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-1.506126882702&{}:{}&2.506126882702&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.285139704438&{}:{}&0.000000000000&{}:{}&-0.285139704438&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.882346458175&{}:{}&0.117653541825&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Hiroyasu Kamo