Derousseau's Generalization of the Malfatti circles

Test Case in ETC

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

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

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4c (310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&, \\ P_{\mathbf{4c}}&{}\approx{}&0.230303806447&{}:{}&1.088725273053&{}:{}&-0.319029079500&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.099334080693&{}:{}&0.927417431241&{}:{}&-0.026751511934&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.349957482996&{}:{}&1.236095791072&{}:{}&-0.586053274068&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.703103932619&{}:{}&1.208559760021&{}:{}&0.494544172598&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.518014890839&{}:{}&1.379419839423&{}:{}&-0.897434730262&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{4c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{4c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{4c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{4c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{4c}}\) Radical center of the Malfatti circles
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&20.865307708125&{}:{}&44.696942343280&{}:{}&-64.562250051405&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.186189011933&{}:{}&1.217220513922&{}:{}&-0.403409525855&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.432633888068&{}:{}&0.648950832103&{}:{}&-0.081584720171&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.260338381621&{}:{}&1.046241756398&{}:{}&-0.306580138020&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.725048476228&{}:{}&1.279327228771&{}:{}&-1.004375704999&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.314597674643&{}:{}&1.487211368808&{}:{}&-0.801809043451&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.136073377978&{}:{}&0.889586906089&{}:{}&-0.025660284068&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.348085546347&{}:{}&0.745656848834&{}:{}&-0.093742395181&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.138081347144&{}:{}&1.289175350263&{}:{}&-0.427256697407&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.709621848012&{}:{}&0.290378151988&,\\B^*_{\mathbf{4c}}&{}\approx{}&3.371234856368&{}:{}&0.000000000000&{}:{}&-2.371234856368&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.391029432250&{}:{}&2.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&9.932653854062&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.062063003978&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.144211296023&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&3.000000000000&{}:{}&4.500000000000&{}:{}&-6.500000000000&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&20.865307708125&{}:{}&44.696942343280&{}:{}&-64.562250051405&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.186189011933&{}:{}&1.217220513922&{}:{}&-0.403409525855&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.432633888068&{}:{}&0.648950832103&{}:{}&-0.081584720171&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.230303806447&{}:{}&1.088725273053&{}:{}&-0.319029079500&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.260338381621&{}:{}&1.046241756398&{}:{}&-0.306580138020&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.725048476228&{}:{}&1.279327228771&{}:{}&-1.004375704999&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.314597674643&{}:{}&1.487211368808&{}:{}&-0.801809043451&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.099334080693&{}:{}&0.927417431241&{}:{}&-0.026751511934&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.136073377978&{}:{}&0.889586906089&{}:{}&-0.025660284068&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.348085546347&{}:{}&0.745656848834&{}:{}&-0.093742395181&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.138081347144&{}:{}&1.289175350263&{}:{}&-0.427256697407&. \end{alignedat} \]
4c (310)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.349957482996&{}:{}&1.236095791072&{}:{}&-0.586053274068&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&20.865307708125&{}:{}&44.696942343280&{}:{}&-64.562250051405&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.186189011933&{}:{}&1.217220513922&{}:{}&-0.403409525855&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.432633888068&{}:{}&0.648950832103&{}:{}&-0.081584720171&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.260338381621&{}:{}&1.046241756398&{}:{}&-0.306580138020&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.725048476228&{}:{}&1.279327228771&{}:{}&-1.004375704999&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.314597674643&{}:{}&1.487211368808&{}:{}&-0.801809043451&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.703103932619&{}:{}&1.208559760021&{}:{}&0.494544172598&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.709621848012&{}:{}&0.290378151988&,\\B^*_{\mathbf{4c}}&{}\approx{}&3.371234856368&{}:{}&0.000000000000&{}:{}&-2.371234856368&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.391029432250&{}:{}&2.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.518014890839&{}:{}&1.379419839423&{}:{}&-0.897434730262&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.260338381621&{}:{}&1.046241756398&{}:{}&-0.306580138020&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.725048476228&{}:{}&1.279327228771&{}:{}&-1.004375704999&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.314597674643&{}:{}&1.487211368808&{}:{}&-0.801809043451&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.709621848012&{}:{}&0.290378151988&,\\B^*_{\mathbf{4c}}&{}\approx{}&3.371234856368&{}:{}&0.000000000000&{}:{}&-2.371234856368&,\\C^*_{\mathbf{4c}}&{}\approx{}&-1.391029432250&{}:{}&2.391029432250&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)