Derousseau's Generalization of the Malfatti circles

Martin's solution

Problem 4331 (proposed by A. Martin) I. Solution by the Proposer, Mathematical Questions with their Solutions, from the “Educational Times”.

\(a:b:c=231:250:289\).


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3(022)

Malfatti circles

3 (022)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&, \\ P_{\mathbf{3}}&{}\approx{}&0.002508551881&{}:{}&0.712493891513&{}:{}&0.284997556605&, \\ P^-_{\mathbf{3}}&{}\approx{}&-0.149827586207&{}:{}&0.911083743842&{}:{}&0.238743842365&, \\ P^+_{\mathbf{3}}&{}\approx{}&0.077768313458&{}:{}&0.614383061572&{}:{}&0.307848624970&, \\ Q_{\mathbf{3}}&{}\approx{}&-0.032786885246&{}:{}&0.590163934426&{}:{}&0.442622950820&, \\ I^\prime_{\mathbf{3}}&{}\approx{}&0.032835820896&{}:{}&0.575692963753&{}:{}&0.391471215352&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{3}}\)
\(P^-_{\mathbf{3}}\)
\(P^+_{\mathbf{3}}\)
\(Q_{\mathbf{3}}\)
\(I^\prime_{\mathbf{3}}\)
3 (022)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3}}&{}\approx{}&-0.718181818182&{}:{}&0.796930342385&{}:{}&0.921251475797&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.122222222222&{}:{}&0.724867724868&{}:{}&0.152910052910&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.275000000000&{}:{}&0.297619047619&{}:{}&0.427380952381&, \\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.169230769231&{}:{}&0.593406593407&{}:{}&0.237362637363&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.004943820225&{}:{}&0.433386837881&{}:{}&0.561669341894&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002588235294&{}:{}&0.735126050420&{}:{}&0.262285714286&, \\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.117857142857&{}:{}&0.698979591837&{}:{}&0.183163265306&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.587162162162&{}:{}&0.651544401544&{}:{}&0.935617760618&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.157142857143&{}:{}&0.955567036941&{}:{}&0.201575820201&, \\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.571428571429&{}:{}&0.428571428571&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.080000000000&{}:{}&0.000000000000&{}:{}&1.080000000000&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.058823529412&{}:{}&1.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3}}}{B^\prime_{\mathbf{3}}}{C^\prime_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3}}}{B^{\prime\prime}_{\mathbf{3}}}{C^{\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3}}}{B^{\prime\prime\prime}_{\mathbf{3}}}{C^{\prime\prime\prime}_{\mathbf{3}}}\)
\(\triangle{A^*_{\mathbf{3}}}{B^*_{\mathbf{3}}}{C^*_{\mathbf{3}}}\)
3 (022)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3}}}}&{}\approx{}&2.454545454545&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3}}}}&{}\approx{}&0.407407407407&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3}}}}&{}\approx{}&0.916666666667&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.718181818182&{}:{}&0.796930342385&{}:{}&0.921251475797&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.122222222222&{}:{}&0.724867724868&{}:{}&0.152910052910&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.275000000000&{}:{}&0.297619047619&{}:{}&0.427380952381&. \end{alignedat} \]
3 (022)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3}}&{}\approx{}&0.002508551881&{}:{}&0.712493891513&{}:{}&0.284997556605&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.169230769231&{}:{}&0.593406593407&{}:{}&0.237362637363&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.004943820225&{}:{}&0.433386837881&{}:{}&0.561669341894&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002588235294&{}:{}&0.735126050420&{}:{}&0.262285714286&. \end{alignedat} \]
3 (022)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3}}&{}\approx{}&-0.149827586207&{}:{}&0.911083743842&{}:{}&0.238743842365&,\\ A^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&0.117857142857&{}:{}&0.698979591837&{}:{}&0.183163265306&,\\B^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.587162162162&{}:{}&0.651544401544&{}:{}&0.935617760618&,\\C^{\prime\prime\prime}_{\mathbf{3}}&{}\approx{}&-0.157142857143&{}:{}&0.955567036941&{}:{}&0.201575820201&. \end{alignedat} \]
3 (022)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3}}&{}\approx{}&0.077768313458&{}:{}&0.614383061572&{}:{}&0.307848624970&,\\ A^\prime_{\mathbf{3}}&{}\approx{}&-0.718181818182&{}:{}&0.796930342385&{}:{}&0.921251475797&,\\B^\prime_{\mathbf{3}}&{}\approx{}&0.122222222222&{}:{}&0.724867724868&{}:{}&0.152910052910&,\\C^\prime_{\mathbf{3}}&{}\approx{}&0.275000000000&{}:{}&0.297619047619&{}:{}&0.427380952381&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.169230769231&{}:{}&0.593406593407&{}:{}&0.237362637363&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.004943820225&{}:{}&0.433386837881&{}:{}&0.561669341894&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002588235294&{}:{}&0.735126050420&{}:{}&0.262285714286&, \end{alignedat} \]
3 (022)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3}}&{}\approx{}&-0.032786885246&{}:{}&0.590163934426&{}:{}&0.442622950820&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.571428571429&{}:{}&0.428571428571&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.080000000000&{}:{}&0.000000000000&{}:{}&1.080000000000&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.058823529412&{}:{}&1.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3}}&{}\approx{}&0.032835820896&{}:{}&0.575692963753&{}:{}&0.391471215352&,\\ A^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.169230769231&{}:{}&0.593406593407&{}:{}&0.237362637363&,\\B^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.004943820225&{}:{}&0.433386837881&{}:{}&0.561669341894&,\\C^{\prime\prime}_{\mathbf{3}}&{}\approx{}&0.002588235294&{}:{}&0.735126050420&{}:{}&0.262285714286&,\\ A^*_{\mathbf{3}}&{}\approx{}&0.000000000000&{}:{}&0.571428571429&{}:{}&0.428571428571&,\\B^*_{\mathbf{3}}&{}\approx{}&-0.080000000000&{}:{}&0.000000000000&{}:{}&1.080000000000&,\\C^*_{\mathbf{3}}&{}\approx{}&-0.058823529412&{}:{}&1.058823529412&{}:{}&0.000000000000&. \end{alignedat} \]
3 (022)