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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\(\mathbf{2c}\) \((130)\)

Malfatti circles

2c (130)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&, \\ P_{\mathbf{2c}}&{}\approx{}&1.060561207988&{}:{}&0.574326049362&{}:{}&-0.634887257350&, \\ P^-_{\mathbf{2c}}&{}\approx{}&0.913743541908&{}:{}&0.277148180383&{}:{}&-0.190891722290&, \\ P^+_{\mathbf{2c}}&{}\approx{}&1.188073265858&{}:{}&0.832426880403&{}:{}&-1.020500146262&, \\ Q_{\mathbf{2c}}&{}\approx{}&1.135573336979&{}:{}&-0.660642613977&{}:{}&0.525069276997&, \\ I^\prime_{\mathbf{2c}}&{}\approx{}&1.311650453174&{}:{}&1.182156927546&{}:{}&-1.493807380720&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{2c}}\)
\(P^-_{\mathbf{2c}}\)
\(P^+_{\mathbf{2c}}\)
\(Q_{\mathbf{2c}}\)
\(I^\prime_{\mathbf{2c}}\)
2c (130)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2c}}&{}\approx{}&1.198601612729&{}:{}&0.446853628640&{}:{}&-0.645455241369&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&18.623725976367&{}:{}&22.727680305761&{}:{}&-40.351406282128&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.602609334218&{}:{}&0.903914001327&{}:{}&-0.506523335545&, \\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.167443605865&{}:{}&1.587934386408&{}:{}&-1.755377992273&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.001438676347&{}:{}&0.598055864696&{}:{}&-0.599494541043&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.472943048489&{}:{}&0.797643319030&{}:{}&-1.270586367519&, \\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.724736667707&{}:{}&0.884440810095&{}:{}&-0.609177477802&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.859179777535&{}:{}&0.320312935648&{}:{}&-0.179492713183&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.307524912015&{}:{}&0.396586277823&{}:{}&-0.704111189838&, \\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.872953846944&{}:{}&-3.872953846944&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.683815606936&{}:{}&0.000000000000&{}:{}&0.316184393064&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.391029432250&{}:{}&-1.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2c}}}{B^\prime_{\mathbf{2c}}}{C^\prime_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2c}}}{B^{\prime\prime\prime}_{\mathbf{2c}}}{C^{\prime\prime\prime}_{\mathbf{2c}}}\)
\(\triangle{A^*_{\mathbf{2c}}}{B^*_{\mathbf{2c}}}{C^*_{\mathbf{2c}}}\)
2c (130)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2c}}}}&{}\approx{}&0.099300806364&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2c}}}}&{}\approx{}&6.207908658789&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2c}}}}&{}\approx{}&0.200869778073&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.198601612729&{}:{}&0.446853628640&{}:{}&-0.645455241369&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&18.623725976367&{}:{}&22.727680305761&{}:{}&-40.351406282128&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.602609334218&{}:{}&0.903914001327&{}:{}&-0.506523335545&. \end{alignedat} \]
2c (130)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2c}}&{}\approx{}&1.060561207988&{}:{}&0.574326049362&{}:{}&-0.634887257350&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.167443605865&{}:{}&1.587934386408&{}:{}&-1.755377992273&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.001438676347&{}:{}&0.598055864696&{}:{}&-0.599494541043&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.472943048489&{}:{}&0.797643319030&{}:{}&-1.270586367519&. \end{alignedat} \]
2c (130)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2c}}&{}\approx{}&0.913743541908&{}:{}&0.277148180383&{}:{}&-0.190891722290&,\\ A^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.724736667707&{}:{}&0.884440810095&{}:{}&-0.609177477802&,\\B^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&0.859179777535&{}:{}&0.320312935648&{}:{}&-0.179492713183&,\\C^{\prime\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.307524912015&{}:{}&0.396586277823&{}:{}&-0.704111189838&. \end{alignedat} \]
2c (130)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2c}}&{}\approx{}&1.188073265858&{}:{}&0.832426880403&{}:{}&-1.020500146262&,\\ A^\prime_{\mathbf{2c}}&{}\approx{}&1.198601612729&{}:{}&0.446853628640&{}:{}&-0.645455241369&,\\B^\prime_{\mathbf{2c}}&{}\approx{}&18.623725976367&{}:{}&22.727680305761&{}:{}&-40.351406282128&,\\C^\prime_{\mathbf{2c}}&{}\approx{}&0.602609334218&{}:{}&0.903914001327&{}:{}&-0.506523335545&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.167443605865&{}:{}&1.587934386408&{}:{}&-1.755377992273&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.001438676347&{}:{}&0.598055864696&{}:{}&-0.599494541043&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.472943048489&{}:{}&0.797643319030&{}:{}&-1.270586367519&, \end{alignedat} \]
2c (130)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2c}}&{}\approx{}&1.135573336979&{}:{}&-0.660642613977&{}:{}&0.525069276997&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.872953846944&{}:{}&-3.872953846944&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.683815606936&{}:{}&0.000000000000&{}:{}&0.316184393064&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.391029432250&{}:{}&-1.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2c}}&{}\approx{}&1.311650453174&{}:{}&1.182156927546&{}:{}&-1.493807380720&,\\ A^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.167443605865&{}:{}&1.587934386408&{}:{}&-1.755377992273&,\\B^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.001438676347&{}:{}&0.598055864696&{}:{}&-0.599494541043&,\\C^{\prime\prime}_{\mathbf{2c}}&{}\approx{}&1.472943048489&{}:{}&0.797643319030&{}:{}&-1.270586367519&,\\ A^*_{\mathbf{2c}}&{}\approx{}&0.000000000000&{}:{}&4.872953846944&{}:{}&-3.872953846944&,\\B^*_{\mathbf{2c}}&{}\approx{}&0.683815606936&{}:{}&0.000000000000&{}:{}&0.316184393064&,\\C^*_{\mathbf{2c}}&{}\approx{}&2.391029432250&{}:{}&-1.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
2c (130)

Hiroyasu Kamo