Derousseau's Generalization of the Malfatti circles

Ajima's example

不朽算法 十四 (Fukyū Sanpō §14)

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


[Top] > Ajima's example > 5c (312)

5c(312)

Malfatti circles

5c (312)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&, \\ P_{\mathbf{5c}}&{}\approx{}&-0.175293056808&{}:{}&-0.077321911632&{}:{}&1.252614968440&, \\ P^-_{\mathbf{5c}}&{}\approx{}&-0.052986348123&{}:{}&0.094816552901&{}:{}&0.958169795222&, \\ P^+_{\mathbf{5c}}&{}\approx{}&-0.312332695985&{}:{}&-0.270195984704&{}:{}&1.582528680688&, \\ Q_{\mathbf{5c}}&{}\approx{}&-0.931034482759&{}:{}&0.482758620690&{}:{}&1.448275862069&, \\ I^\prime_{\mathbf{5c}}&{}\approx{}&-0.504672897196&{}:{}&-0.408878504673&{}:{}&1.913551401869&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{5c}}\)
\(P^-_{\mathbf{5c}}\)
\(P^+_{\mathbf{5c}}\)
\(Q_{\mathbf{5c}}\)
\(I^\prime_{\mathbf{5c}}\)
5c (312)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5c}}&{}\approx{}&-8.625000000000&{}:{}&-27.343750000000&{}:{}&36.968750000000&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.540000000000&{}:{}&0.453571428571&{}:{}&1.086428571429&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.166666666667&{}:{}&-0.248015873016&{}:{}&1.414682539683&, \\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.254716981132&{}:{}&-0.082547169811&{}:{}&1.337264150943&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.242696629213&{}:{}&-0.491573033708&{}:{}&1.734269662921&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.770816812054&{}:{}&-0.340007930214&{}:{}&2.110824742268&, \\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.120076238882&{}:{}&0.100857687421&{}:{}&1.019218551461&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.071875000000&{}:{}&-0.227864583333&{}:{}&1.299739583333&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.197017766497&{}:{}&0.352553934010&{}:{}&0.844463832487&, \\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.250000000000&{}:{}&0.750000000000&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.800000000000&{}:{}&0.000000000000&{}:{}&2.800000000000&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.076923076923&{}:{}&-1.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5c}}}{B^\prime_{\mathbf{5c}}}{C^\prime_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5c}}}{B^{\prime\prime\prime}_{\mathbf{5c}}}{C^{\prime\prime\prime}_{\mathbf{5c}}}\)
\(\triangle{A^*_{\mathbf{5c}}}{B^*_{\mathbf{5c}}}{C^*_{\mathbf{5c}}}\)
5c (312)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5c}}}}&{}\approx{}&-8.750000000000&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.257142857143&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5c}}}}&{}\approx{}&-0.079365079365&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&3.125000000000&{}:{}&-4.225000000000&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-8.625000000000&{}:{}&-27.343750000000&{}:{}&36.968750000000&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.540000000000&{}:{}&0.453571428571&{}:{}&1.086428571429&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.166666666667&{}:{}&-0.248015873016&{}:{}&1.414682539683&. \end{alignedat} \]
5c (312)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5c}}&{}\approx{}&-0.175293056808&{}:{}&-0.077321911632&{}:{}&1.252614968440&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.254716981132&{}:{}&-0.082547169811&{}:{}&1.337264150943&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.242696629213&{}:{}&-0.491573033708&{}:{}&1.734269662921&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.770816812054&{}:{}&-0.340007930214&{}:{}&2.110824742268&. \end{alignedat} \]
5c (312)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5c}}&{}\approx{}&-0.052986348123&{}:{}&0.094816552901&{}:{}&0.958169795222&,\\ A^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.120076238882&{}:{}&0.100857687421&{}:{}&1.019218551461&,\\B^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.071875000000&{}:{}&-0.227864583333&{}:{}&1.299739583333&,\\C^{\prime\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.197017766497&{}:{}&0.352553934010&{}:{}&0.844463832487&. \end{alignedat} \]
5c (312)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5c}}&{}\approx{}&-0.312332695985&{}:{}&-0.270195984704&{}:{}&1.582528680688&,\\ A^\prime_{\mathbf{5c}}&{}\approx{}&-8.625000000000&{}:{}&-27.343750000000&{}:{}&36.968750000000&,\\B^\prime_{\mathbf{5c}}&{}\approx{}&-0.540000000000&{}:{}&0.453571428571&{}:{}&1.086428571429&,\\C^\prime_{\mathbf{5c}}&{}\approx{}&-0.166666666667&{}:{}&-0.248015873016&{}:{}&1.414682539683&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.254716981132&{}:{}&-0.082547169811&{}:{}&1.337264150943&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.242696629213&{}:{}&-0.491573033708&{}:{}&1.734269662921&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.770816812054&{}:{}&-0.340007930214&{}:{}&2.110824742268&, \end{alignedat} \]
5c (312)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5c}}&{}\approx{}&-0.931034482759&{}:{}&0.482758620690&{}:{}&1.448275862069&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.250000000000&{}:{}&0.750000000000&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.800000000000&{}:{}&0.000000000000&{}:{}&2.800000000000&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.076923076923&{}:{}&-1.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5c}}&{}\approx{}&-0.504672897196&{}:{}&-0.408878504673&{}:{}&1.913551401869&,\\ A^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.254716981132&{}:{}&-0.082547169811&{}:{}&1.337264150943&,\\B^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.242696629213&{}:{}&-0.491573033708&{}:{}&1.734269662921&,\\C^{\prime\prime}_{\mathbf{5c}}&{}\approx{}&-0.770816812054&{}:{}&-0.340007930214&{}:{}&2.110824742268&,\\ A^*_{\mathbf{5c}}&{}\approx{}&0.000000000000&{}:{}&0.250000000000&{}:{}&0.750000000000&,\\B^*_{\mathbf{5c}}&{}\approx{}&-1.800000000000&{}:{}&0.000000000000&{}:{}&2.800000000000&,\\C^*_{\mathbf{5c}}&{}\approx{}&2.076923076923&{}:{}&-1.076923076923&{}:{}&0.000000000000&. \end{alignedat} \]
5c (312)