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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5b (303)

Malfatti circles

5b (303)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&, \\ P_{\mathbf{5b}}&{}\approx{}&0.034066133721&{}:{}&-0.007789789244&{}:{}&0.973723655523&, \\ P^-_{\mathbf{5b}}&{}\approx{}&0.000717840735&{}:{}&0.044135241194&{}:{}&0.955146918070&, \\ P^+_{\mathbf{5b}}&{}\approx{}&0.064185684647&{}:{}&-0.054687500000&{}:{}&0.990501815353&, \\ Q_{\mathbf{5b}}&{}\approx{}&0.286624203822&{}:{}&0.044585987261&{}:{}&0.668789808917&, \\ I^\prime_{\mathbf{5b}}&{}\approx{}&0.125000000000&{}:{}&-0.072916666667&{}:{}&0.947916666667&, \end{alignedat} \]
\(I_{\mathbf{b}}\) Incenter
\(P_{\mathbf{5b}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{5b}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{5b}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{5b}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{5b}}\) Radical center of the Malfatti circles
5b (303)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5b}}&{}\approx{}&0.037500000000&{}:{}&-2.734375000000&{}:{}&3.696875000000&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.234375000000&{}:{}&0.294084821429&{}:{}&0.471540178571&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.037500000000&{}:{}&-0.055803571429&{}:{}&1.018303571429&, \\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.064655172414&{}:{}&-0.007543103448&{}:{}&0.942887931034&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.037500000000&{}:{}&-0.109375000000&{}:{}&1.071875000000&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.212104072398&{}:{}&-0.048501131222&{}:{}&0.836397058824&, \\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.034002590674&{}:{}&0.042665155440&{}:{}&0.923332253886&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.000794491525&{}:{}&-0.057931673729&{}:{}&1.057137182203&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.006208609272&{}:{}&0.381725993377&{}:{}&0.612065397351&, \\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.062500000000&{}:{}&0.937500000000&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.300000000000&{}:{}&0.000000000000&{}:{}&0.700000000000&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.865384615385&{}:{}&0.134615384615&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5b}}}{B^\prime_{\mathbf{5b}}}{C^\prime_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^*_{\mathbf{5b}}}{B^*_{\mathbf{5b}}}{C^*_{\mathbf{5b}}}\)
5b (303)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5b}}}}&{}\approx{}&2.800000000000&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5b}}}}&{}\approx{}&0.357142857143&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5b}}}}&{}\approx{}&0.057142857143&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.656250000000&{}:{}&-0.976562500000&{}:{}&1.320312500000&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.037500000000&{}:{}&-2.734375000000&{}:{}&3.696875000000&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.234375000000&{}:{}&0.294084821429&{}:{}&0.471540178571&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.037500000000&{}:{}&-0.055803571429&{}:{}&1.018303571429&. \end{alignedat} \]
5b (303)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5b}}&{}\approx{}&0.034066133721&{}:{}&-0.007789789244&{}:{}&0.973723655523&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.064655172414&{}:{}&-0.007543103448&{}:{}&0.942887931034&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.037500000000&{}:{}&-0.109375000000&{}:{}&1.071875000000&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.212104072398&{}:{}&-0.048501131222&{}:{}&0.836397058824&. \end{alignedat} \]
5b (303)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5b}}&{}\approx{}&0.000717840735&{}:{}&0.044135241194&{}:{}&0.955146918070&,\\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.034002590674&{}:{}&0.042665155440&{}:{}&0.923332253886&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.000794491525&{}:{}&-0.057931673729&{}:{}&1.057137182203&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.006208609272&{}:{}&0.381725993377&{}:{}&0.612065397351&. \end{alignedat} \]
5b (303)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5b}}&{}\approx{}&0.064185684647&{}:{}&-0.054687500000&{}:{}&0.990501815353&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&0.037500000000&{}:{}&-2.734375000000&{}:{}&3.696875000000&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.234375000000&{}:{}&0.294084821429&{}:{}&0.471540178571&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.037500000000&{}:{}&-0.055803571429&{}:{}&1.018303571429&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.064655172414&{}:{}&-0.007543103448&{}:{}&0.942887931034&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.037500000000&{}:{}&-0.109375000000&{}:{}&1.071875000000&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.212104072398&{}:{}&-0.048501131222&{}:{}&0.836397058824&, \end{alignedat} \]
5b (303)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5b}}&{}\approx{}&0.286624203822&{}:{}&0.044585987261&{}:{}&0.668789808917&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.062500000000&{}:{}&0.937500000000&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.300000000000&{}:{}&0.000000000000&{}:{}&0.700000000000&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.865384615385&{}:{}&0.134615384615&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5b}}&{}\approx{}&0.125000000000&{}:{}&-0.072916666667&{}:{}&0.947916666667&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.064655172414&{}:{}&-0.007543103448&{}:{}&0.942887931034&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.037500000000&{}:{}&-0.109375000000&{}:{}&1.071875000000&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.212104072398&{}:{}&-0.048501131222&{}:{}&0.836397058824&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.062500000000&{}:{}&0.937500000000&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.300000000000&{}:{}&0.000000000000&{}:{}&0.700000000000&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.865384615385&{}:{}&0.134615384615&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)