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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\(\mathbf{5a}\) \((213)\)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&, \\ P_{\mathbf{5a}}&{}\approx{}&1.055378837526&{}:{}&-0.000423763529&{}:{}&-0.054955073997&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.852549193227&{}:{}&0.079399583775&{}:{}&0.068051222998&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.338503357919&{}:{}&-0.111847056705&{}:{}&-0.226656301214&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.442713608994&{}:{}&0.108168733268&{}:{}&-0.550882342262&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.563551983312&{}:{}&-0.038372098607&{}:{}&-0.525179884705&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{5a}}\)
\(P^-_{\mathbf{5a}}\)
\(P^+_{\mathbf{5a}}\)
\(Q_{\mathbf{5a}}\)
\(I^\prime_{\mathbf{5a}}\)
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.296510114500&{}:{}&-0.121299592295&{}:{}&-0.175210522204&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&3.062841355180&{}:{}&4.573314914377&{}:{}&-6.636156269557&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.282466322424&{}:{}&-1.923699483636&{}:{}&1.641233161212&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.807923019671&{}:{}&-0.006182294991&{}:{}&-0.801740724680&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.295677103731&{}:{}&-0.228209357951&{}:{}&-0.067467745780&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.341945902724&{}:{}&-0.000538828060&{}:{}&-0.341407074664&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.265047512049&{}:{}&0.395758578165&{}:{}&0.339193909786&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.013929015791&{}:{}&-0.094861717511&{}:{}&0.080932701720&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.043850210556&{}:{}&0.097215823908&{}:{}&-0.141066034464&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.244331168209&{}:{}&1.244331168209&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.617697946698&{}:{}&0.000000000000&{}:{}&-0.617697946698&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.930253423925&{}:{}&0.069746576075&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.215643719636&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-8.167576947147&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-3.419910193131&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.375000000000&{}:{}&0.562500000000&{}:{}&0.812500000000&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.296510114500&{}:{}&-0.121299592295&{}:{}&-0.175210522204&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&3.062841355180&{}:{}&4.573314914377&{}:{}&-6.636156269557&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.282466322424&{}:{}&-1.923699483636&{}:{}&1.641233161212&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.055378837526&{}:{}&-0.000423763529&{}:{}&-0.054955073997&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.807923019671&{}:{}&-0.006182294991&{}:{}&-0.801740724680&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.295677103731&{}:{}&-0.228209357951&{}:{}&-0.067467745780&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.341945902724&{}:{}&-0.000538828060&{}:{}&-0.341407074664&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.852549193227&{}:{}&0.079399583775&{}:{}&0.068051222998&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.265047512049&{}:{}&0.395758578165&{}:{}&0.339193909786&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.013929015791&{}:{}&-0.094861717511&{}:{}&0.080932701720&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.043850210556&{}:{}&0.097215823908&{}:{}&-0.141066034464&. \end{alignedat} \]
5a (213)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.338503357919&{}:{}&-0.111847056705&{}:{}&-0.226656301214&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.296510114500&{}:{}&-0.121299592295&{}:{}&-0.175210522204&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&3.062841355180&{}:{}&4.573314914377&{}:{}&-6.636156269557&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.282466322424&{}:{}&-1.923699483636&{}:{}&1.641233161212&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.807923019671&{}:{}&-0.006182294991&{}:{}&-0.801740724680&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.295677103731&{}:{}&-0.228209357951&{}:{}&-0.067467745780&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.341945902724&{}:{}&-0.000538828060&{}:{}&-0.341407074664&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.442713608994&{}:{}&0.108168733268&{}:{}&-0.550882342262&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.244331168209&{}:{}&1.244331168209&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.617697946698&{}:{}&0.000000000000&{}:{}&-0.617697946698&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.930253423925&{}:{}&0.069746576075&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.563551983312&{}:{}&-0.038372098607&{}:{}&-0.525179884705&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.807923019671&{}:{}&-0.006182294991&{}:{}&-0.801740724680&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.295677103731&{}:{}&-0.228209357951&{}:{}&-0.067467745780&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.341945902724&{}:{}&-0.000538828060&{}:{}&-0.341407074664&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.244331168209&{}:{}&1.244331168209&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.617697946698&{}:{}&0.000000000000&{}:{}&-0.617697946698&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.930253423925&{}:{}&0.069746576075&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Hiroyasu Kamo