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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2 (020)

Malfatti circles

2 (020)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&, \\ P_{\mathbf{2}}&{}\approx{}&0.001021152443&{}:{}&0.995866763919&{}:{}&0.003112083637&, \\ P^-_{\mathbf{2}}&{}\approx{}&-0.007420869302&{}:{}&1.021252965824&{}:{}&-0.013832096522&, \\ P^+_{\mathbf{2}}&{}\approx{}&0.008864499789&{}:{}&0.972280849866&{}:{}&0.018854650345&, \\ Q_{\mathbf{2}}&{}\approx{}&-0.028571428571&{}:{}&1.142857142857&{}:{}&-0.114285714286&, \\ I^\prime_{\mathbf{2}}&{}\approx{}&0.024054982818&{}:{}&0.916380297824&{}:{}&0.059564719359&, \end{alignedat} \]
\(I\) Incenter
\(P_{\mathbf{2}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{2}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{2}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{2}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{2}}\) Radical center of the Malfatti circles
2 (020)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2}}&{}\approx{}&-0.333333333333&{}:{}&0.566893424036&{}:{}&0.766439909297&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.008101851852&{}:{}&0.975597993827&{}:{}&0.016300154321&,\\C^\prime_{\mathbf{2}}&{}\approx{}&1.166666666667&{}:{}&1.736111111111&{}:{}&-1.902777777778&, \\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.016091954023&{}:{}&0.980842911877&{}:{}&0.003065134100&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.035897435897&{}:{}&0.854700854701&{}:{}&0.109401709402&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000991641876&{}:{}&0.967086933938&{}:{}&0.031921424187&, \\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.008348240906&{}:{}&1.005267342477&{}:{}&-0.013615583383&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.859649122807&{}:{}&-1.461988304094&{}:{}&1.602339181287&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.007198472161&{}:{}&0.990646883111&{}:{}&0.016551589050&, \\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.111111111111&{}:{}&-0.111111111111&,\\B^*_{\mathbf{2}}&{}\approx{}&0.200000000000&{}:{}&0.000000000000&{}:{}&0.800000000000&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.025641025641&{}:{}&1.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2}}}{B^\prime_{\mathbf{2}}}{C^\prime_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2}}}{B^{\prime\prime}_{\mathbf{2}}}{C^{\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2}}}{B^{\prime\prime\prime}_{\mathbf{2}}}{C^{\prime\prime\prime}_{\mathbf{2}}}\)
\(\triangle{A^*_{\mathbf{2}}}{B^*_{\mathbf{2}}}{C^*_{\mathbf{2}}}\)
2 (020)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2}}}}&{}\approx{}&1.714285714286&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2}}}}&{}\approx{}&0.036458333333&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2}}}}&{}\approx{}&5.250000000000&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.333333333333&{}:{}&0.566893424036&{}:{}&0.766439909297&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.008101851852&{}:{}&0.975597993827&{}:{}&0.016300154321&,\\C^\prime_{\mathbf{2}}&{}\approx{}&1.166666666667&{}:{}&1.736111111111&{}:{}&-1.902777777778&. \end{alignedat} \]
2 (020)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2}}&{}\approx{}&0.001021152443&{}:{}&0.995866763919&{}:{}&0.003112083637&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.016091954023&{}:{}&0.980842911877&{}:{}&0.003065134100&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.035897435897&{}:{}&0.854700854701&{}:{}&0.109401709402&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000991641876&{}:{}&0.967086933938&{}:{}&0.031921424187&. \end{alignedat} \]
2 (020)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2}}&{}\approx{}&-0.007420869302&{}:{}&1.021252965824&{}:{}&-0.013832096522&,\\ A^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.008348240906&{}:{}&1.005267342477&{}:{}&-0.013615583383&,\\B^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&0.859649122807&{}:{}&-1.461988304094&{}:{}&1.602339181287&,\\C^{\prime\prime\prime}_{\mathbf{2}}&{}\approx{}&-0.007198472161&{}:{}&0.990646883111&{}:{}&0.016551589050&. \end{alignedat} \]
2 (020)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2}}&{}\approx{}&0.008864499789&{}:{}&0.972280849866&{}:{}&0.018854650345&,\\ A^\prime_{\mathbf{2}}&{}\approx{}&-0.333333333333&{}:{}&0.566893424036&{}:{}&0.766439909297&,\\B^\prime_{\mathbf{2}}&{}\approx{}&0.008101851852&{}:{}&0.975597993827&{}:{}&0.016300154321&,\\C^\prime_{\mathbf{2}}&{}\approx{}&1.166666666667&{}:{}&1.736111111111&{}:{}&-1.902777777778&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.016091954023&{}:{}&0.980842911877&{}:{}&0.003065134100&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.035897435897&{}:{}&0.854700854701&{}:{}&0.109401709402&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000991641876&{}:{}&0.967086933938&{}:{}&0.031921424187&, \end{alignedat} \]
2 (020)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2}}&{}\approx{}&-0.028571428571&{}:{}&1.142857142857&{}:{}&-0.114285714286&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.111111111111&{}:{}&-0.111111111111&,\\B^*_{\mathbf{2}}&{}\approx{}&0.200000000000&{}:{}&0.000000000000&{}:{}&0.800000000000&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.025641025641&{}:{}&1.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2}}&{}\approx{}&0.024054982818&{}:{}&0.916380297824&{}:{}&0.059564719359&,\\ A^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.016091954023&{}:{}&0.980842911877&{}:{}&0.003065134100&,\\B^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.035897435897&{}:{}&0.854700854701&{}:{}&0.109401709402&,\\C^{\prime\prime}_{\mathbf{2}}&{}\approx{}&0.000991641876&{}:{}&0.967086933938&{}:{}&0.031921424187&,\\ A^*_{\mathbf{2}}&{}\approx{}&0.000000000000&{}:{}&1.111111111111&{}:{}&-0.111111111111&,\\B^*_{\mathbf{2}}&{}\approx{}&0.200000000000&{}:{}&0.000000000000&{}:{}&0.800000000000&,\\C^*_{\mathbf{2}}&{}\approx{}&-0.025641025641&{}:{}&1.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
2 (020)