Derousseau's Generalization of the Malfatti circles

Half of an Equilateral Triangle

\(A=30\degree\), \(B=60\degree\), \(C=90\degree\).


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\(\mathbf{5a}\) \((213)\)

Malfatti circles

5a (213)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&, \\ P_{\mathbf{5a}}&{}\approx{}&1.028766514796&{}:{}&-0.000919487004&{}:{}&-0.027847027791&, \\ P^-_{\mathbf{5a}}&{}\approx{}&0.891684398963&{}:{}&0.061478798430&{}:{}&0.046836802607&, \\ P^+_{\mathbf{5a}}&{}\approx{}&1.199386096640&{}:{}&-0.078583661532&{}:{}&-0.120802435108&, \\ Q_{\mathbf{5a}}&{}\approx{}&1.318961298407&{}:{}&0.100253727199&{}:{}&-0.419215025605&, \\ I^\prime_{\mathbf{5a}}&{}\approx{}&1.373661241130&{}:{}&-0.054047342660&{}:{}&-0.319613898470&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{5a}}\)
\(P^-_{\mathbf{5a}}\)
\(P^+_{\mathbf{5a}}\)
\(Q_{\mathbf{5a}}\)
\(I^\prime_{\mathbf{5a}}\)
5a (213)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5a}}&{}\approx{}&1.177250691694&{}:{}&-0.082262332300&{}:{}&-0.094988359395&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.410431731523&{}:{}&2.410431731523&{}:{}&-2.820863463046&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.754600960884&{}:{}&-3.039058011260&{}:{}&2.284457050376&, \\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.563803670862&{}:{}&-0.018021305398&{}:{}&-0.545782365463&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.192466745950&{}:{}&-0.160188620509&{}:{}&-0.032278125442&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.184846922835&{}:{}&-0.001058987955&{}:{}&-0.183787934880&, \\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.249315396705&{}:{}&0.426080702769&{}:{}&0.324603900526&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.017656790442&{}:{}&-0.071110445424&{}:{}&0.053453654982&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.011879249019&{}:{}&0.069765850404&{}:{}&-0.081645099424&, \\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.314313139868&{}:{}&1.314313139868&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.929359733804&{}:{}&0.070640266196&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5a}}}{B^\prime_{\mathbf{5a}}}{C^\prime_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5a}}}{B^{\prime\prime\prime}_{\mathbf{5a}}}{C^{\prime\prime\prime}_{\mathbf{5a}}}\)
\(\triangle{A^*_{\mathbf{5a}}}{B^*_{\mathbf{5a}}}{C^*_{\mathbf{5a}}}\)
5a (213)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5a}}}}&{}\approx{}&-0.129756511997&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5a}}}}&{}\approx{}&-3.853371151129&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5a}}}}&{}\approx{}&-4.793658972144&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.366025403784&{}:{}&0.633974596216&{}:{}&0.732050807569&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.177250691694&{}:{}&-0.082262332300&{}:{}&-0.094988359395&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.410431731523&{}:{}&2.410431731523&{}:{}&-2.820863463046&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.754600960884&{}:{}&-3.039058011260&{}:{}&2.284457050376&. \end{alignedat} \]
5a (213)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5a}}&{}\approx{}&1.028766514796&{}:{}&-0.000919487004&{}:{}&-0.027847027791&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.563803670862&{}:{}&-0.018021305398&{}:{}&-0.545782365463&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.192466745950&{}:{}&-0.160188620509&{}:{}&-0.032278125442&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.184846922835&{}:{}&-0.001058987955&{}:{}&-0.183787934880&. \end{alignedat} \]
5a (213)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5a}}&{}\approx{}&0.891684398963&{}:{}&0.061478798430&{}:{}&0.046836802607&,\\ A^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&0.249315396705&{}:{}&0.426080702769&{}:{}&0.324603900526&,\\B^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.017656790442&{}:{}&-0.071110445424&{}:{}&0.053453654982&,\\C^{\prime\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.011879249019&{}:{}&0.069765850404&{}:{}&-0.081645099424&. \end{alignedat} \]
5a (213)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5a}}&{}\approx{}&1.199386096640&{}:{}&-0.078583661532&{}:{}&-0.120802435108&,\\ A^\prime_{\mathbf{5a}}&{}\approx{}&1.177250691694&{}:{}&-0.082262332300&{}:{}&-0.094988359395&,\\B^\prime_{\mathbf{5a}}&{}\approx{}&1.410431731523&{}:{}&2.410431731523&{}:{}&-2.820863463046&,\\C^\prime_{\mathbf{5a}}&{}\approx{}&1.754600960884&{}:{}&-3.039058011260&{}:{}&2.284457050376&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.563803670862&{}:{}&-0.018021305398&{}:{}&-0.545782365463&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.192466745950&{}:{}&-0.160188620509&{}:{}&-0.032278125442&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.184846922835&{}:{}&-0.001058987955&{}:{}&-0.183787934880&, \end{alignedat} \]
5a (213)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5a}}&{}\approx{}&1.318961298407&{}:{}&0.100253727199&{}:{}&-0.419215025605&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.314313139868&{}:{}&1.314313139868&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.929359733804&{}:{}&0.070640266196&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5a}}&{}\approx{}&1.373661241130&{}:{}&-0.054047342660&{}:{}&-0.319613898470&,\\ A^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.563803670862&{}:{}&-0.018021305398&{}:{}&-0.545782365463&,\\B^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.192466745950&{}:{}&-0.160188620509&{}:{}&-0.032278125442&,\\C^{\prime\prime}_{\mathbf{5a}}&{}\approx{}&1.184846922835&{}:{}&-0.001058987955&{}:{}&-0.183787934880&,\\ A^*_{\mathbf{5a}}&{}\approx{}&0.000000000000&{}:{}&-0.314313139868&{}:{}&1.314313139868&,\\B^*_{\mathbf{5a}}&{}\approx{}&1.465925826289&{}:{}&0.000000000000&{}:{}&-0.465925826289&,\\C^*_{\mathbf{5a}}&{}\approx{}&0.929359733804&{}:{}&0.070640266196&{}:{}&0.000000000000&. \end{alignedat} \]
5a (213)

Hiroyasu Kamo