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”.

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

[Top] > Martin's solution > 0c (110)

0c (110)

Malfatti circles

0c (110)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&, \\ P_{\mathbf{c}}&{}\approx{}&0.811942475713&{}:{}&0.823386998228&{}:{}&-0.635329473941&, \\ P^-_{\mathbf{c}}&{}\approx{}&0.725220665074&{}:{}&0.717264126297&{}:{}&-0.442484791371&, \\ P^+_{\mathbf{c}}&{}\approx{}&0.872024809950&{}:{}&0.896910715209&{}:{}&-0.768935525159&, \\ Q_{\mathbf{c}}&{}\approx{}&0.704697986577&{}:{}&0.664429530201&{}:{}&-0.369127516779&, \\ I^\prime_{\mathbf{c}}&{}\approx{}&0.945000000000&{}:{}&0.990000000000&{}:{}&-0.935000000000&, \end{alignedat} \]
\(I_{\mathbf{c}}\) Incenter
\(P_{\mathbf{c}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{c}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{c}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{c}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{c}}\) Radical center of the Malfatti circles
0c (110)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{c}}&{}\approx{}&1.066203703704&{}:{}&0.424382716049&{}:{}&-0.490586419753&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.410156250000&{}:{}&1.102982954545&{}:{}&-0.513139204545&,\\C^\prime_{\mathbf{c}}&{}\approx{}&2.362500000000&{}:{}&2.556818181818&{}:{}&-3.919318181818&, \\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.698964497041&{}:{}&1.318047337278&{}:{}&-1.017011834320&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.250735294118&{}:{}&0.727941176471&{}:{}&-0.978676470588&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.745550366069&{}:{}&0.756059075991&{}:{}&-0.501609442060&, \\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.492560652009&{}:{}&1.324583017437&{}:{}&-0.817143669447&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.269207217505&{}:{}&0.505184519922&{}:{}&-0.774391737427&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.654063162678&{}:{}&0.646887306877&{}:{}&-0.300950469555&, \\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&2.250000000000&{}:{}&-1.250000000000&,\\B^*_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&0.000000000000&{}:{}&-1.100000000000&,\\C^*_{\mathbf{c}}&{}\approx{}&0.514705882353&{}:{}&0.485294117647&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{c}}}{B^\prime_{\mathbf{c}}}{C^\prime_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{c}}}{B^{\prime\prime}_{\mathbf{c}}}{C^{\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{c}}}{B^{\prime\prime\prime}_{\mathbf{c}}}{C^{\prime\prime\prime}_{\mathbf{c}}}\)
\(\triangle{A^*_{\mathbf{c}}}{B^*_{\mathbf{c}}}{C^*_{\mathbf{c}}}\)
0c (110)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{c}}}}&{}\approx{}&0.325925925926&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{c}}}}&{}\approx{}&0.340909090909&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{c}}}}&{}\approx{}&1.963636363636&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.203125000000&{}:{}&1.302083333333&{}:{}&-1.505208333333&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.066203703704&{}:{}&0.424382716049&{}:{}&-0.490586419753&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.410156250000&{}:{}&1.102982954545&{}:{}&-0.513139204545&,\\C^\prime_{\mathbf{c}}&{}\approx{}&2.362500000000&{}:{}&2.556818181818&{}:{}&-3.919318181818&. \end{alignedat} \]
0c (110)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{c}}&{}\approx{}&0.811942475713&{}:{}&0.823386998228&{}:{}&-0.635329473941&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.698964497041&{}:{}&1.318047337278&{}:{}&-1.017011834320&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.250735294118&{}:{}&0.727941176471&{}:{}&-0.978676470588&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.745550366069&{}:{}&0.756059075991&{}:{}&-0.501609442060&. \end{alignedat} \]
0c (110)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{c}}&{}\approx{}&0.725220665074&{}:{}&0.717264126297&{}:{}&-0.442484791371&,\\ A^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.492560652009&{}:{}&1.324583017437&{}:{}&-0.817143669447&,\\B^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&1.269207217505&{}:{}&0.505184519922&{}:{}&-0.774391737427&,\\C^{\prime\prime\prime}_{\mathbf{c}}&{}\approx{}&0.654063162678&{}:{}&0.646887306877&{}:{}&-0.300950469555&. \end{alignedat} \]
0c (110)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{c}}&{}\approx{}&0.872024809950&{}:{}&0.896910715209&{}:{}&-0.768935525159&,\\ A^\prime_{\mathbf{c}}&{}\approx{}&1.066203703704&{}:{}&0.424382716049&{}:{}&-0.490586419753&,\\B^\prime_{\mathbf{c}}&{}\approx{}&0.410156250000&{}:{}&1.102982954545&{}:{}&-0.513139204545&,\\C^\prime_{\mathbf{c}}&{}\approx{}&2.362500000000&{}:{}&2.556818181818&{}:{}&-3.919318181818&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.698964497041&{}:{}&1.318047337278&{}:{}&-1.017011834320&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.250735294118&{}:{}&0.727941176471&{}:{}&-0.978676470588&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.745550366069&{}:{}&0.756059075991&{}:{}&-0.501609442060&, \end{alignedat} \]
0c (110)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{c}}&{}\approx{}&0.704697986577&{}:{}&0.664429530201&{}:{}&-0.369127516779&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&2.250000000000&{}:{}&-1.250000000000&,\\B^*_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&0.000000000000&{}:{}&-1.100000000000&,\\C^*_{\mathbf{c}}&{}\approx{}&0.514705882353&{}:{}&0.485294117647&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{c}}&{}\approx{}&0.945000000000&{}:{}&0.990000000000&{}:{}&-0.935000000000&,\\ A^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.698964497041&{}:{}&1.318047337278&{}:{}&-1.017011834320&,\\B^{\prime\prime}_{\mathbf{c}}&{}\approx{}&1.250735294118&{}:{}&0.727941176471&{}:{}&-0.978676470588&,\\C^{\prime\prime}_{\mathbf{c}}&{}\approx{}&0.745550366069&{}:{}&0.756059075991&{}:{}&-0.501609442060&,\\ A^*_{\mathbf{c}}&{}\approx{}&0.000000000000&{}:{}&2.250000000000&{}:{}&-1.250000000000&,\\B^*_{\mathbf{c}}&{}\approx{}&2.100000000000&{}:{}&0.000000000000&{}:{}&-1.100000000000&,\\C^*_{\mathbf{c}}&{}\approx{}&0.514705882353&{}:{}&0.485294117647&{}:{}&0.000000000000&. \end{alignedat} \]
0c (110)