Derousseau's Generalization of the Malfatti circles

Ajima's example

不朽算法 十四 (Fukyū Sanpō §14)

\(a=252\),  \(b=375\),  \(c=507\).  \(a:b:c=84:125:169\).


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7(222)

Malfatti circles

7 (222)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&, \\ P_{\mathbf{7}}&{}\approx{}&0.800000000000&{}:{}&0.190476190476&{}:{}&0.009523809524&, \\ P^-_{\mathbf{7}}&{}\approx{}&-0.500000000000&{}:{}&0.505952380952&{}:{}&0.994047619048&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.428571428571&{}:{}&0.280612244898&{}:{}&0.290816326531&, \\ Q_{\mathbf{7}}&{}\approx{}&0.533333333333&{}:{}&0.333333333333&{}:{}&0.133333333333&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.571428571429&{}:{}&0.340136054422&{}:{}&0.088435374150&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&-0.505952380952&{}:{}&1.005952380952&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.000000000000&{}:{}&2.976190476190&{}:{}&-3.976190476190&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.800000000000&{}:{}&0.190476190476&{}:{}&0.009523809524&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.400000000000&{}:{}&0.595238095238&{}:{}&0.004761904762&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.307692307692&{}:{}&0.073260073260&{}:{}&0.619047619048&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.500000000000&{}:{}&0.505952380952&{}:{}&0.994047619048&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&2.000000000000&{}:{}&2.976190476191&{}:{}&-3.976190476191&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&-0.505952380952&{}:{}&1.005952380952&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.714285714286&{}:{}&0.285714285714&,\\B^*_{\mathbf{7}}&{}\approx{}&0.800000000000&{}:{}&0.000000000000&{}:{}&0.200000000000&,\\C^*_{\mathbf{7}}&{}\approx{}&0.615384615385&{}:{}&0.384615384615&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&1.000000000000&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&2.250000000000&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&9.000000000000&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&-0.505952380952&{}:{}&1.005952380952&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.000000000000&{}:{}&2.976190476190&{}:{}&-3.976190476190&. \end{alignedat} \]
7 (222)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.800000000000&{}:{}&0.190476190476&{}:{}&0.009523809524&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.800000000000&{}:{}&0.190476190476&{}:{}&0.009523809524&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.400000000000&{}:{}&0.595238095238&{}:{}&0.004761904762&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.307692307692&{}:{}&0.073260073260&{}:{}&0.619047619048&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&-0.500000000000&{}:{}&0.505952380952&{}:{}&0.994047619048&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-0.500000000000&{}:{}&0.505952380952&{}:{}&0.994047619048&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&2.000000000000&{}:{}&2.976190476191&{}:{}&-3.976190476191&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&-0.505952380952&{}:{}&1.005952380952&. \end{alignedat} \]
7 (222)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.428571428571&{}:{}&0.280612244898&{}:{}&0.290816326531&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.500000000000&{}:{}&-0.505952380952&{}:{}&1.005952380952&,\\C^\prime_{\mathbf{7}}&{}\approx{}&2.000000000000&{}:{}&2.976190476190&{}:{}&-3.976190476190&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.800000000000&{}:{}&0.190476190476&{}:{}&0.009523809524&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.400000000000&{}:{}&0.595238095238&{}:{}&0.004761904762&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.307692307692&{}:{}&0.073260073260&{}:{}&0.619047619048&, \end{alignedat} \]
7 (222)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.533333333333&{}:{}&0.333333333333&{}:{}&0.133333333333&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.714285714286&{}:{}&0.285714285714&,\\B^*_{\mathbf{7}}&{}\approx{}&0.800000000000&{}:{}&0.000000000000&{}:{}&0.200000000000&,\\C^*_{\mathbf{7}}&{}\approx{}&0.615384615385&{}:{}&0.384615384615&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.571428571429&{}:{}&0.340136054422&{}:{}&0.088435374150&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.800000000000&{}:{}&0.190476190476&{}:{}&0.009523809524&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.400000000000&{}:{}&0.595238095238&{}:{}&0.004761904762&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.307692307692&{}:{}&0.073260073260&{}:{}&0.619047619048&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.714285714286&{}:{}&0.285714285714&,\\B^*_{\mathbf{7}}&{}\approx{}&0.800000000000&{}:{}&0.000000000000&{}:{}&0.200000000000&,\\C^*_{\mathbf{7}}&{}\approx{}&0.615384615385&{}:{}&0.384615384615&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)