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 > 0a (011)

0a(011)

Malfatti circles

0a (011)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&, \\ P_{\mathbf{a}}&{}\approx{}&-0.226347305389&{}:{}&0.681304058550&{}:{}&0.545043246840&, \\ P^-_{\mathbf{a}}&{}\approx{}&-0.176349614396&{}:{}&0.706083976007&{}:{}&0.470265638389&, \\ P^+_{\mathbf{a}}&{}\approx{}&-0.258075040783&{}:{}&0.665579119086&{}:{}&0.592495921697&, \\ Q_{\mathbf{a}}&{}\approx{}&-0.142857142857&{}:{}&0.761904761905&{}:{}&0.380952380952&, \\ I^\prime_{\mathbf{a}}&{}\approx{}&-0.301435406699&{}:{}&0.637958532695&{}:{}&0.663476874003&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{a}}\)
\(P^-_{\mathbf{a}}\)
\(P^+_{\mathbf{a}}\)
\(Q_{\mathbf{a}}\)
\(I^\prime_{\mathbf{a}}\)
0a (011)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{a}}&{}\approx{}&-0.777777777778&{}:{}&0.755857898715&{}:{}&1.021919879063&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.140000000000&{}:{}&0.858333333333&{}:{}&0.281666666667&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.218750000000&{}:{}&0.325520833333&{}:{}&0.893229166667&, \\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.170731707317&{}:{}&0.650406504065&{}:{}&0.520325203252&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.387030716724&{}:{}&0.455062571104&{}:{}&0.931968145620&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.277804997550&{}:{}&0.836191409440&{}:{}&0.441613588110&, \\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.108527131783&{}:{}&0.665374677003&{}:{}&0.443152454780&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.379005524862&{}:{}&0.368324125230&{}:{}&1.010681399632&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.231600270088&{}:{}&0.927301372946&{}:{}&0.304298897142&, \\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.666666666667&{}:{}&0.333333333333&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.600000000000&{}:{}&0.000000000000&{}:{}&1.600000000000&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.230769230769&{}:{}&1.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{a}}}{B^\prime_{\mathbf{a}}}{C^\prime_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{a}}}{B^{\prime\prime}_{\mathbf{a}}}{C^{\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{a}}}{B^{\prime\prime\prime}_{\mathbf{a}}}{C^{\prime\prime\prime}_{\mathbf{a}}}\)
\(\triangle{A^*_{\mathbf{a}}}{B^*_{\mathbf{a}}}{C^*_{\mathbf{a}}}\)
0a (011)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{a}}}}&{}\approx{}&1.269841269841&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{a}}}}&{}\approx{}&0.350000000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{a}}}}&{}\approx{}&0.546875000000&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.400000000000&{}:{}&0.595238095238&{}:{}&0.804761904762&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.777777777778&{}:{}&0.755857898715&{}:{}&1.021919879063&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.140000000000&{}:{}&0.858333333333&{}:{}&0.281666666667&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.218750000000&{}:{}&0.325520833333&{}:{}&0.893229166667&. \end{alignedat} \]
0a (011)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{a}}&{}\approx{}&-0.226347305389&{}:{}&0.681304058550&{}:{}&0.545043246840&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.170731707317&{}:{}&0.650406504065&{}:{}&0.520325203252&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.387030716724&{}:{}&0.455062571104&{}:{}&0.931968145620&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.277804997550&{}:{}&0.836191409440&{}:{}&0.441613588110&. \end{alignedat} \]
0a (011)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{a}}&{}\approx{}&-0.176349614396&{}:{}&0.706083976007&{}:{}&0.470265638389&,\\ A^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.108527131783&{}:{}&0.665374677003&{}:{}&0.443152454780&,\\B^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.379005524862&{}:{}&0.368324125230&{}:{}&1.010681399632&,\\C^{\prime\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.231600270088&{}:{}&0.927301372946&{}:{}&0.304298897142&. \end{alignedat} \]
0a (011)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{a}}&{}\approx{}&-0.258075040783&{}:{}&0.665579119086&{}:{}&0.592495921697&,\\ A^\prime_{\mathbf{a}}&{}\approx{}&-0.777777777778&{}:{}&0.755857898715&{}:{}&1.021919879063&,\\B^\prime_{\mathbf{a}}&{}\approx{}&-0.140000000000&{}:{}&0.858333333333&{}:{}&0.281666666667&,\\C^\prime_{\mathbf{a}}&{}\approx{}&-0.218750000000&{}:{}&0.325520833333&{}:{}&0.893229166667&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.170731707317&{}:{}&0.650406504065&{}:{}&0.520325203252&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.387030716724&{}:{}&0.455062571104&{}:{}&0.931968145620&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.277804997550&{}:{}&0.836191409440&{}:{}&0.441613588110&, \end{alignedat} \]
0a (011)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{a}}&{}\approx{}&-0.142857142857&{}:{}&0.761904761905&{}:{}&0.380952380952&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.666666666667&{}:{}&0.333333333333&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.600000000000&{}:{}&0.000000000000&{}:{}&1.600000000000&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.230769230769&{}:{}&1.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{a}}&{}\approx{}&-0.301435406699&{}:{}&0.637958532695&{}:{}&0.663476874003&,\\ A^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.170731707317&{}:{}&0.650406504065&{}:{}&0.520325203252&,\\B^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.387030716724&{}:{}&0.455062571104&{}:{}&0.931968145620&,\\C^{\prime\prime}_{\mathbf{a}}&{}\approx{}&-0.277804997550&{}:{}&0.836191409440&{}:{}&0.441613588110&,\\ A^*_{\mathbf{a}}&{}\approx{}&0.000000000000&{}:{}&0.666666666667&{}:{}&0.333333333333&,\\B^*_{\mathbf{a}}&{}\approx{}&-0.600000000000&{}:{}&0.000000000000&{}:{}&1.600000000000&,\\C^*_{\mathbf{a}}&{}\approx{}&-0.230769230769&{}:{}&1.230769230769&{}:{}&0.000000000000&. \end{alignedat} \]
0a (011)