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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6c(330)

Malfatti circles

6c (330)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&, \\ P_{\mathbf{6c}}&{}\approx{}&0.358878504673&{}:{}&0.801401869159&{}:{}&-0.160280373832&, \\ P^-_{\mathbf{6c}}&{}\approx{}&0.237000000000&{}:{}&0.638750000000&{}:{}&0.124250000000&, \\ P^+_{\mathbf{6c}}&{}\approx{}&0.465789473684&{}:{}&0.944078947368&{}:{}&-0.409868421053&, \\ Q_{\mathbf{6c}}&{}\approx{}&0.391304347826&{}:{}&0.456521739130&{}:{}&0.152173913043&, \\ I^\prime_{\mathbf{6c}}&{}\approx{}&0.533333333333&{}:{}&0.972222222222&{}:{}&-0.505555555556&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{6c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{6c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{6c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{6c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{6c}}\) Radical center of the Malfatti circles
6c (330)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{6c}}&{}\approx{}&1.481250000000&{}:{}&1.367187500000&{}:{}&-1.848437500000&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.300000000000&{}:{}&1.303571428571&{}:{}&-0.603571428571&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.300000000000&{}:{}&0.446428571429&{}:{}&0.253571428571&, \\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.300000000000&{}:{}&0.875000000000&{}:{}&-0.175000000000&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.590769230769&{}:{}&0.673076923077&{}:{}&-0.263846153846&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.590769230769&{}:{}&1.319230769231&{}:{}&-0.910000000000&, \\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.161538461538&{}:{}&0.701923076923&{}:{}&0.136538461538&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.408620689655&{}:{}&0.377155172414&{}:{}&0.214224137931&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.408620689655&{}:{}&1.101293103448&{}:{}&-0.509913793103&, \\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.750000000000&{}:{}&0.250000000000&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.720000000000&{}:{}&0.000000000000&{}:{}&0.280000000000&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.461538461538&{}:{}&0.538461538462&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{6c}}}{B^\prime_{\mathbf{6c}}}{C^\prime_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{6c}}}{B^{\prime\prime\prime}_{\mathbf{6c}}}{C^{\prime\prime\prime}_{\mathbf{6c}}}\)
\(\triangle{A^*_{\mathbf{6c}}}{B^*_{\mathbf{6c}}}{C^*_{\mathbf{6c}}}\)
6c (330)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{6c}}}}&{}\approx{}&0.437500000000&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{6c}}}}&{}\approx{}&0.142857142857&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{6c}}}}&{}\approx{}&0.142857142857&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.481250000000&{}:{}&1.367187500000&{}:{}&-1.848437500000&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.300000000000&{}:{}&1.303571428571&{}:{}&-0.603571428571&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.300000000000&{}:{}&0.446428571429&{}:{}&0.253571428571&. \end{alignedat} \]
6c (330)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{6c}}&{}\approx{}&0.358878504673&{}:{}&0.801401869159&{}:{}&-0.160280373832&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.300000000000&{}:{}&0.875000000000&{}:{}&-0.175000000000&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.590769230769&{}:{}&0.673076923077&{}:{}&-0.263846153846&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.590769230769&{}:{}&1.319230769231&{}:{}&-0.910000000000&. \end{alignedat} \]
6c (330)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{6c}}&{}\approx{}&0.237000000000&{}:{}&0.638750000000&{}:{}&0.124250000000&,\\ A^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.161538461538&{}:{}&0.701923076923&{}:{}&0.136538461538&,\\B^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.408620689655&{}:{}&0.377155172414&{}:{}&0.214224137931&,\\C^{\prime\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.408620689655&{}:{}&1.101293103448&{}:{}&-0.509913793103&. \end{alignedat} \]
6c (330)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{6c}}&{}\approx{}&0.465789473684&{}:{}&0.944078947368&{}:{}&-0.409868421053&,\\ A^\prime_{\mathbf{6c}}&{}\approx{}&1.481250000000&{}:{}&1.367187500000&{}:{}&-1.848437500000&,\\B^\prime_{\mathbf{6c}}&{}\approx{}&0.300000000000&{}:{}&1.303571428571&{}:{}&-0.603571428571&,\\C^\prime_{\mathbf{6c}}&{}\approx{}&0.300000000000&{}:{}&0.446428571429&{}:{}&0.253571428571&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.300000000000&{}:{}&0.875000000000&{}:{}&-0.175000000000&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.590769230769&{}:{}&0.673076923077&{}:{}&-0.263846153846&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.590769230769&{}:{}&1.319230769231&{}:{}&-0.910000000000&, \end{alignedat} \]
6c (330)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{6c}}&{}\approx{}&0.391304347826&{}:{}&0.456521739130&{}:{}&0.152173913043&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.750000000000&{}:{}&0.250000000000&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.720000000000&{}:{}&0.000000000000&{}:{}&0.280000000000&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.461538461538&{}:{}&0.538461538462&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{6c}}&{}\approx{}&0.533333333333&{}:{}&0.972222222222&{}:{}&-0.505555555556&,\\ A^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.300000000000&{}:{}&0.875000000000&{}:{}&-0.175000000000&,\\B^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.590769230769&{}:{}&0.673076923077&{}:{}&-0.263846153846&,\\C^{\prime\prime}_{\mathbf{6c}}&{}\approx{}&0.590769230769&{}:{}&1.319230769231&{}:{}&-0.910000000000&,\\ A^*_{\mathbf{6c}}&{}\approx{}&0.000000000000&{}:{}&0.750000000000&{}:{}&0.250000000000&,\\B^*_{\mathbf{6c}}&{}\approx{}&0.720000000000&{}:{}&0.000000000000&{}:{}&0.280000000000&,\\C^*_{\mathbf{6c}}&{}\approx{}&0.461538461538&{}:{}&0.538461538462&{}:{}&0.000000000000&. \end{alignedat} \]
6c (330)