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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7c (332)

Malfatti circles

7c (332)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&, \\ P_{\mathbf{7c}}&{}\approx{}&-0.057731958763&{}:{}&-0.069587628866&{}:{}&1.127319587629&, \\ P^-_{\mathbf{7c}}&{}\approx{}&0.007000000000&{}:{}&0.026250000000&{}:{}&0.966750000000&, \\ P^+_{\mathbf{7c}}&{}\approx{}&-0.126595744681&{}:{}&-0.171542553191&{}:{}&1.298138297872&, \\ Q_{\mathbf{7c}}&{}\approx{}&0.225806451613&{}:{}&0.193548387097&{}:{}&0.580645161290&, \\ I^\prime_{\mathbf{7c}}&{}\approx{}&-0.254545454545&{}:{}&-0.340909090909&{}:{}&1.595454545455&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{7c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{7c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{7c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{7c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{7c}}\) Radical center of the Malfatti circles
7c (332)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7c}}&{}\approx{}&0.175000000000&{}:{}&-2.343750000000&{}:{}&3.168750000000&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.700000000000&{}:{}&0.291666666667&{}:{}&1.408333333333&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.077777777778&{}:{}&-0.115740740741&{}:{}&1.193518518519&, \\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.140000000000&{}:{}&-0.075000000000&{}:{}&1.215000000000&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.065882352941&{}:{}&-0.220588235294&{}:{}&1.286470588235&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.430769230769&{}:{}&-0.519230769231&{}:{}&1.950000000000&, \\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.067741935484&{}:{}&0.028225806452&{}:{}&1.039516129032&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.007954545455&{}:{}&-0.106534090909&{}:{}&1.098579545455&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.043750000000&{}:{}&0.164062500000&{}:{}&0.792187500000&, \\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.250000000000&{}:{}&0.750000000000&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.280000000000&{}:{}&0.000000000000&{}:{}&0.720000000000&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.538461538462&{}:{}&0.461538461538&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7c}}}{B^\prime_{\mathbf{7c}}}{C^\prime_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7c}}}{B^{\prime\prime\prime}_{\mathbf{7c}}}{C^{\prime\prime\prime}_{\mathbf{7c}}}\)
\(\triangle{A^*_{\mathbf{7c}}}{B^*_{\mathbf{7c}}}{C^*_{\mathbf{7c}}}\)
7c (332)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.750000000000&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.333333333333&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7c}}}}&{}\approx{}&-0.037037037037&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.175000000000&{}:{}&-2.343750000000&{}:{}&3.168750000000&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.700000000000&{}:{}&0.291666666667&{}:{}&1.408333333333&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.077777777778&{}:{}&-0.115740740741&{}:{}&1.193518518519&. \end{alignedat} \]
7c (332)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7c}}&{}\approx{}&-0.057731958763&{}:{}&-0.069587628866&{}:{}&1.127319587629&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.140000000000&{}:{}&-0.075000000000&{}:{}&1.215000000000&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.065882352941&{}:{}&-0.220588235294&{}:{}&1.286470588235&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.430769230769&{}:{}&-0.519230769231&{}:{}&1.950000000000&. \end{alignedat} \]
7c (332)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7c}}&{}\approx{}&0.007000000000&{}:{}&0.026250000000&{}:{}&0.966750000000&,\\ A^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.067741935484&{}:{}&0.028225806452&{}:{}&1.039516129032&,\\B^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.007954545455&{}:{}&-0.106534090909&{}:{}&1.098579545455&,\\C^{\prime\prime\prime}_{\mathbf{7c}}&{}\approx{}&0.043750000000&{}:{}&0.164062500000&{}:{}&0.792187500000&. \end{alignedat} \]
7c (332)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7c}}&{}\approx{}&-0.126595744681&{}:{}&-0.171542553191&{}:{}&1.298138297872&,\\ A^\prime_{\mathbf{7c}}&{}\approx{}&0.175000000000&{}:{}&-2.343750000000&{}:{}&3.168750000000&,\\B^\prime_{\mathbf{7c}}&{}\approx{}&-0.700000000000&{}:{}&0.291666666667&{}:{}&1.408333333333&,\\C^\prime_{\mathbf{7c}}&{}\approx{}&-0.077777777778&{}:{}&-0.115740740741&{}:{}&1.193518518519&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.140000000000&{}:{}&-0.075000000000&{}:{}&1.215000000000&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.065882352941&{}:{}&-0.220588235294&{}:{}&1.286470588235&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.430769230769&{}:{}&-0.519230769231&{}:{}&1.950000000000&, \end{alignedat} \]
7c (332)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7c}}&{}\approx{}&0.225806451613&{}:{}&0.193548387097&{}:{}&0.580645161290&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.250000000000&{}:{}&0.750000000000&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.280000000000&{}:{}&0.000000000000&{}:{}&0.720000000000&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.538461538462&{}:{}&0.461538461538&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7c}}&{}\approx{}&-0.254545454545&{}:{}&-0.340909090909&{}:{}&1.595454545455&,\\ A^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.140000000000&{}:{}&-0.075000000000&{}:{}&1.215000000000&,\\B^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.065882352941&{}:{}&-0.220588235294&{}:{}&1.286470588235&,\\C^{\prime\prime}_{\mathbf{7c}}&{}\approx{}&-0.430769230769&{}:{}&-0.519230769231&{}:{}&1.950000000000&,\\ A^*_{\mathbf{7c}}&{}\approx{}&0.000000000000&{}:{}&0.250000000000&{}:{}&0.750000000000&,\\B^*_{\mathbf{7c}}&{}\approx{}&0.280000000000&{}:{}&0.000000000000&{}:{}&0.720000000000&,\\C^*_{\mathbf{7c}}&{}\approx{}&0.538461538462&{}:{}&0.461538461538&{}:{}&0.000000000000&. \end{alignedat} \]
7c (332)