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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\(\mathbf{4a}\) \((211)\)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&, \\ P_{\mathbf{4a}}&{}\approx{}&1.056620232005&{}:{}&-0.001600516836&{}:{}&-0.055019715170&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.708490426358&{}:{}&0.135572875173&{}:{}&0.155936698470&, \\ P^+_{\mathbf{4a}}&{}\approx{}&1.734368889050&{}:{}&-0.268653439934&{}:{}&-0.465715449116&, \\ Q_{\mathbf{4a}}&{}\approx{}&2.116232844801&{}:{}&-0.308175515164&{}:{}&-0.808057329637&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.622206882543&{}:{}&-0.077325463419&{}:{}&-0.544881419124&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&1.893705898413&{}:{}&-0.365606958442&{}:{}&-0.528098939971&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&3.124513520549&{}:{}&4.645265773973&{}:{}&-6.769779294522&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.826425905618&{}:{}&-1.239638858427&{}:{}&1.413212952809&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.307121490516&{}:{}&-0.008681580748&{}:{}&-0.298439909768&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.650624752914&{}:{}&-0.564674379254&{}:{}&-0.085950373660&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.982771265040&{}:{}&-0.003003405287&{}:{}&-0.979767859754&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.238279829948&{}:{}&0.354254552399&{}:{}&0.407465617652&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.973678701061&{}:{}&-0.187982573584&{}:{}&0.214303872523&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.095909697900&{}:{}&0.209707320165&{}:{}&-0.305617018065&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.276085331657&{}:{}&0.723914668343&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.617697946698&{}:{}&0.000000000000&{}:{}&-0.617697946698&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.170445654633&{}:{}&-0.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-0.649967926118&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-8.332036054796&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-2.203802414982&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&1.893705898413&{}:{}&-0.365606958442&{}:{}&-0.528098939971&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&3.124513520549&{}:{}&4.645265773973&{}:{}&-6.769779294522&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.826425905618&{}:{}&-1.239638858427&{}:{}&1.413212952809&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.056620232005&{}:{}&-0.001600516836&{}:{}&-0.055019715170&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.307121490516&{}:{}&-0.008681580748&{}:{}&-0.298439909768&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.650624752914&{}:{}&-0.564674379254&{}:{}&-0.085950373660&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.982771265040&{}:{}&-0.003003405287&{}:{}&-0.979767859754&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.708490426358&{}:{}&0.135572875173&{}:{}&0.155936698470&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.238279829948&{}:{}&0.354254552399&{}:{}&0.407465617652&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.973678701061&{}:{}&-0.187982573584&{}:{}&0.214303872523&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.095909697900&{}:{}&0.209707320165&{}:{}&-0.305617018065&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&1.734368889050&{}:{}&-0.268653439934&{}:{}&-0.465715449116&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&1.893705898413&{}:{}&-0.365606958442&{}:{}&-0.528098939971&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&3.124513520549&{}:{}&4.645265773973&{}:{}&-6.769779294522&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.826425905618&{}:{}&-1.239638858427&{}:{}&1.413212952809&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.307121490516&{}:{}&-0.008681580748&{}:{}&-0.298439909768&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.650624752914&{}:{}&-0.564674379254&{}:{}&-0.085950373660&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.982771265040&{}:{}&-0.003003405287&{}:{}&-0.979767859754&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&2.116232844801&{}:{}&-0.308175515164&{}:{}&-0.808057329637&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.276085331657&{}:{}&0.723914668343&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.617697946698&{}:{}&0.000000000000&{}:{}&-0.617697946698&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.170445654633&{}:{}&-0.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.622206882543&{}:{}&-0.077325463419&{}:{}&-0.544881419124&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.307121490516&{}:{}&-0.008681580748&{}:{}&-0.298439909768&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.650624752914&{}:{}&-0.564674379254&{}:{}&-0.085950373660&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.982771265040&{}:{}&-0.003003405287&{}:{}&-0.979767859754&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.276085331657&{}:{}&0.723914668343&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.617697946698&{}:{}&0.000000000000&{}:{}&-0.617697946698&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.170445654633&{}:{}&-0.170445654633&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Hiroyasu Kamo