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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7(222)

Malfatti circles

7 (222)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&, \\ P_{\mathbf{7}}&{}\approx{}&0.507692307692&{}:{}&0.351648351648&{}:{}&0.140659340659&, \\ P^-_{\mathbf{7}}&{}\approx{}&0.212903225806&{}:{}&0.313364055300&{}:{}&0.473732718894&, \\ P^+_{\mathbf{7}}&{}\approx{}&0.347368421053&{}:{}&0.330827067669&{}:{}&0.321804511278&, \\ Q_{\mathbf{7}}&{}\approx{}&0.391304347826&{}:{}&0.347826086957&{}:{}&0.260869565217&, \\ I^\prime_{\mathbf{7}}&{}\approx{}&0.407407407407&{}:{}&0.352733686067&{}:{}&0.239858906526&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{7}}\)
\(P^-_{\mathbf{7}}\)
\(P^+_{\mathbf{7}}\)
\(Q_{\mathbf{7}}\)
\(I^\prime_{\mathbf{7}}\)
7 (222)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7}}&{}\approx{}&-0.272727272727&{}:{}&0.590318772137&{}:{}&0.682408500590&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.660000000000&{}:{}&-0.485714285714&{}:{}&0.825714285714&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.031250000000&{}:{}&1.116071428571&{}:{}&-1.147321428571&, \\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.804878048780&{}:{}&0.139372822300&{}:{}&0.055749128920&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.178378378378&{}:{}&0.772200772201&{}:{}&0.049420849421&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.149321266968&{}:{}&0.103425985779&{}:{}&0.747252747253&, \\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-1.178571428571&{}:{}&0.867346938776&{}:{}&1.311224489796&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.942857142857&{}:{}&-2.040816326531&{}:{}&2.097959183673&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-33.000000000002&{}:{}&-48.571428571431&{}:{}&82.571428571433&, \\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.571428571429&{}:{}&0.428571428571&,\\B^*_{\mathbf{7}}&{}\approx{}&0.600000000000&{}:{}&0.000000000000&{}:{}&0.400000000000&,\\C^*_{\mathbf{7}}&{}\approx{}&0.529411764706&{}:{}&0.470588235294&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7}}}{B^\prime_{\mathbf{7}}}{C^\prime_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7}}}{B^{\prime\prime}_{\mathbf{7}}}{C^{\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7}}}{B^{\prime\prime\prime}_{\mathbf{7}}}{C^{\prime\prime\prime}_{\mathbf{7}}}\)
\(\triangle{A^*_{\mathbf{7}}}{B^*_{\mathbf{7}}}{C^*_{\mathbf{7}}}\)
7 (222)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7}}}}&{}\approx{}&1.818181818182&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7}}}}&{}\approx{}&2.200000000000&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7}}}}&{}\approx{}&3.437500000000&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.300000000000&{}:{}&0.324675324675&{}:{}&0.375324675325&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&-0.272727272727&{}:{}&0.590318772137&{}:{}&0.682408500590&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.660000000000&{}:{}&-0.485714285714&{}:{}&0.825714285714&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.031250000000&{}:{}&1.116071428571&{}:{}&-1.147321428571&. \end{alignedat} \]
7 (222)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7}}&{}\approx{}&0.507692307692&{}:{}&0.351648351648&{}:{}&0.140659340659&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.804878048780&{}:{}&0.139372822300&{}:{}&0.055749128920&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.178378378378&{}:{}&0.772200772201&{}:{}&0.049420849421&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.149321266968&{}:{}&0.103425985779&{}:{}&0.747252747253&. \end{alignedat} \]
7 (222)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7}}&{}\approx{}&0.212903225806&{}:{}&0.313364055300&{}:{}&0.473732718894&,\\ A^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-1.178571428571&{}:{}&0.867346938776&{}:{}&1.311224489796&,\\B^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&0.942857142857&{}:{}&-2.040816326531&{}:{}&2.097959183673&,\\C^{\prime\prime\prime}_{\mathbf{7}}&{}\approx{}&-33.000000000002&{}:{}&-48.571428571431&{}:{}&82.571428571433&. \end{alignedat} \]
7 (222)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7}}&{}\approx{}&0.347368421053&{}:{}&0.330827067669&{}:{}&0.321804511278&,\\ A^\prime_{\mathbf{7}}&{}\approx{}&-0.272727272727&{}:{}&0.590318772137&{}:{}&0.682408500590&,\\B^\prime_{\mathbf{7}}&{}\approx{}&0.660000000000&{}:{}&-0.485714285714&{}:{}&0.825714285714&,\\C^\prime_{\mathbf{7}}&{}\approx{}&1.031250000000&{}:{}&1.116071428571&{}:{}&-1.147321428571&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.804878048780&{}:{}&0.139372822300&{}:{}&0.055749128920&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.178378378378&{}:{}&0.772200772201&{}:{}&0.049420849421&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.149321266968&{}:{}&0.103425985779&{}:{}&0.747252747253&, \end{alignedat} \]
7 (222)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7}}&{}\approx{}&0.391304347826&{}:{}&0.347826086957&{}:{}&0.260869565217&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.571428571429&{}:{}&0.428571428571&,\\B^*_{\mathbf{7}}&{}\approx{}&0.600000000000&{}:{}&0.000000000000&{}:{}&0.400000000000&,\\C^*_{\mathbf{7}}&{}\approx{}&0.529411764706&{}:{}&0.470588235294&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7}}&{}\approx{}&0.407407407407&{}:{}&0.352733686067&{}:{}&0.239858906526&,\\ A^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.804878048780&{}:{}&0.139372822300&{}:{}&0.055749128920&,\\B^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.178378378378&{}:{}&0.772200772201&{}:{}&0.049420849421&,\\C^{\prime\prime}_{\mathbf{7}}&{}\approx{}&0.149321266968&{}:{}&0.103425985779&{}:{}&0.747252747253&,\\ A^*_{\mathbf{7}}&{}\approx{}&0.000000000000&{}:{}&0.571428571429&{}:{}&0.428571428571&,\\B^*_{\mathbf{7}}&{}\approx{}&0.600000000000&{}:{}&0.000000000000&{}:{}&0.400000000000&,\\C^*_{\mathbf{7}}&{}\approx{}&0.529411764706&{}:{}&0.470588235294&{}:{}&0.000000000000&. \end{alignedat} \]
7 (222)