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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7a (233)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&, \\ P_{\mathbf{7a}}&{}\approx{}&1.008839354343&{}:{}&-0.004472084411&{}:{}&-0.004367269932&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.962761976048&{}:{}&0.016909363092&{}:{}&0.020328660860&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.057464454976&{}:{}&-0.027035760448&{}:{}&-0.030428694528&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.803571428571&{}:{}&0.107142857143&{}:{}&0.089285714286&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.166666666667&{}:{}&-0.080808080808&{}:{}&-0.085858585859&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{7a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{7a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{7a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{7a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{7a}}\) Radical center of the Malfatti circles
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.050000000000&{}:{}&-0.023191094620&{}:{}&-0.026808905380&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.640625000000&{}:{}&1.411931818182&{}:{}&-2.052556818182&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&1.050000000000&{}:{}&-1.136363636364&{}:{}&1.086363636364&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.280487804878&{}:{}&-0.141906873614&{}:{}&-0.138580931264&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.050000000000&{}:{}&-0.045454545455&{}:{}&-0.004545454545&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.057614826753&{}:{}&-0.004688301223&{}:{}&-0.052926525529&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.345394736842&{}:{}&0.297248803828&{}:{}&0.357356459330&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.000972762646&{}:{}&-0.022108241953&{}:{}&0.021135479307&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.008032915361&{}:{}&0.017704474209&{}:{}&-0.025737389570&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.545454545455&{}:{}&0.454545454545&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.900000000000&{}:{}&0.000000000000&{}:{}&0.100000000000&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.882352941176&{}:{}&0.117647058824&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.028571428571&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-2.187500000000&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.400000000000&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.750000000000&{}:{}&0.811688311688&{}:{}&0.938311688312&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.050000000000&{}:{}&-0.023191094620&{}:{}&-0.026808905380&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.640625000000&{}:{}&1.411931818182&{}:{}&-2.052556818182&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&1.050000000000&{}:{}&-1.136363636364&{}:{}&1.086363636364&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.008839354343&{}:{}&-0.004472084411&{}:{}&-0.004367269932&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.280487804878&{}:{}&-0.141906873614&{}:{}&-0.138580931264&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.050000000000&{}:{}&-0.045454545455&{}:{}&-0.004545454545&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.057614826753&{}:{}&-0.004688301223&{}:{}&-0.052926525529&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.962761976048&{}:{}&0.016909363092&{}:{}&0.020328660860&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.345394736842&{}:{}&0.297248803828&{}:{}&0.357356459330&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.000972762646&{}:{}&-0.022108241953&{}:{}&0.021135479307&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.008032915361&{}:{}&0.017704474209&{}:{}&-0.025737389570&. \end{alignedat} \]
7a (233)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.057464454976&{}:{}&-0.027035760448&{}:{}&-0.030428694528&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.050000000000&{}:{}&-0.023191094620&{}:{}&-0.026808905380&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.640625000000&{}:{}&1.411931818182&{}:{}&-2.052556818182&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&1.050000000000&{}:{}&-1.136363636364&{}:{}&1.086363636364&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.280487804878&{}:{}&-0.141906873614&{}:{}&-0.138580931264&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.050000000000&{}:{}&-0.045454545455&{}:{}&-0.004545454545&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.057614826753&{}:{}&-0.004688301223&{}:{}&-0.052926525529&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.803571428571&{}:{}&0.107142857143&{}:{}&0.089285714286&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.545454545455&{}:{}&0.454545454545&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.900000000000&{}:{}&0.000000000000&{}:{}&0.100000000000&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.882352941176&{}:{}&0.117647058824&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.166666666667&{}:{}&-0.080808080808&{}:{}&-0.085858585859&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.280487804878&{}:{}&-0.141906873614&{}:{}&-0.138580931264&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.050000000000&{}:{}&-0.045454545455&{}:{}&-0.004545454545&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.057614826753&{}:{}&-0.004688301223&{}:{}&-0.052926525529&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.545454545455&{}:{}&0.454545454545&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.900000000000&{}:{}&0.000000000000&{}:{}&0.100000000000&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.882352941176&{}:{}&0.117647058824&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)