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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4(200)

Malfatti circles

4 (200)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&, \\ P_{\mathbf{4}}&{}\approx{}&0.998876264203&{}:{}&0.000380524291&{}:{}&0.000743211506&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.016732660621&{}:{}&-0.007213717011&{}:{}&-0.009518943610&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.981804860178&{}:{}&0.007640912025&{}:{}&0.010554227797&, \\ Q_{\mathbf{4}}&{}\approx{}&1.095890410959&{}:{}&-0.027397260274&{}:{}&-0.068493150685&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.941176470588&{}:{}&0.022408963585&{}:{}&0.036414565826&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{4}}\)
\(P^-_{\mathbf{4}}\)
\(P^+_{\mathbf{4}}\)
\(Q_{\mathbf{4}}\)
\(I^\prime_{\mathbf{4}}\)
4 (200)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.982716049383&{}:{}&0.007348618460&{}:{}&0.009935332158&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.625000000000&{}:{}&-0.882440476190&{}:{}&1.257440476190&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.600000000000&{}:{}&2.380952380952&{}:{}&-2.980952380952&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.898876404494&{}:{}&0.034242910647&{}:{}&0.066880684858&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984615384615&{}:{}&0.014652014652&{}:{}&0.000732600733&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.979911807937&{}:{}&0.000373299736&{}:{}&0.019714892326&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.439560439560&{}:{}&0.620617477760&{}:{}&0.818942961800&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.001887979862&{}:{}&0.007491983578&{}:{}&-0.009379963439&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.996993987976&{}:{}&-0.007073671152&{}:{}&0.010079683176&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.285714285714&{}:{}&0.714285714286&,\\B^*_{\mathbf{4}}&{}\approx{}&1.066666666667&{}:{}&0.000000000000&{}:{}&-0.066666666667&,\\C^*_{\mathbf{4}}&{}\approx{}&1.025641025641&{}:{}&-0.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
4 (200)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.022222222222&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&2.812500000000&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&7.200000000000&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.222222222222&{}:{}&0.330687830688&{}:{}&0.447089947090&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.982716049383&{}:{}&0.007348618460&{}:{}&0.009935332158&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.625000000000&{}:{}&-0.882440476190&{}:{}&1.257440476190&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.600000000000&{}:{}&2.380952380952&{}:{}&-2.980952380952&. \end{alignedat} \]
4 (200)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.998876264203&{}:{}&0.000380524291&{}:{}&0.000743211506&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.898876404494&{}:{}&0.034242910647&{}:{}&0.066880684858&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984615384615&{}:{}&0.014652014652&{}:{}&0.000732600733&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.979911807937&{}:{}&0.000373299736&{}:{}&0.019714892326&. \end{alignedat} \]
4 (200)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.016732660621&{}:{}&-0.007213717011&{}:{}&-0.009518943610&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.439560439560&{}:{}&0.620617477760&{}:{}&0.818942961800&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.001887979862&{}:{}&0.007491983578&{}:{}&-0.009379963439&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.996993987976&{}:{}&-0.007073671152&{}:{}&0.010079683176&. \end{alignedat} \]
4 (200)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.981804860178&{}:{}&0.007640912025&{}:{}&0.010554227797&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.982716049383&{}:{}&0.007348618460&{}:{}&0.009935332158&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.625000000000&{}:{}&-0.882440476190&{}:{}&1.257440476190&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.600000000000&{}:{}&2.380952380952&{}:{}&-2.980952380952&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.898876404494&{}:{}&0.034242910647&{}:{}&0.066880684858&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984615384615&{}:{}&0.014652014652&{}:{}&0.000732600733&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.979911807937&{}:{}&0.000373299736&{}:{}&0.019714892326&, \end{alignedat} \]
4 (200)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.095890410959&{}:{}&-0.027397260274&{}:{}&-0.068493150685&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.285714285714&{}:{}&0.714285714286&,\\B^*_{\mathbf{4}}&{}\approx{}&1.066666666667&{}:{}&0.000000000000&{}:{}&-0.066666666667&,\\C^*_{\mathbf{4}}&{}\approx{}&1.025641025641&{}:{}&-0.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.941176470588&{}:{}&0.022408963585&{}:{}&0.036414565826&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.898876404494&{}:{}&0.034242910647&{}:{}&0.066880684858&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984615384615&{}:{}&0.014652014652&{}:{}&0.000732600733&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.979911807937&{}:{}&0.000373299736&{}:{}&0.019714892326&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.285714285714&{}:{}&0.714285714286&,\\B^*_{\mathbf{4}}&{}\approx{}&1.066666666667&{}:{}&0.000000000000&{}:{}&-0.066666666667&,\\C^*_{\mathbf{4}}&{}\approx{}&1.025641025641&{}:{}&-0.025641025641&{}:{}&0.000000000000&. \end{alignedat} \]
4 (200)