Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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4c(310)

Malfatti circles

4c (310)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&, \\ P_{\mathbf{4c}}&{}\approx{}&0.058686208501&{}:{}&1.016222916292&{}:{}&-0.074909124793&, \\ P^-_{\mathbf{4c}}&{}\approx{}&0.012995544980&{}:{}&0.940138771310&{}:{}&0.046865683709&, \\ P^+_{\mathbf{4c}}&{}\approx{}&0.101121793394&{}:{}&1.086886700633&{}:{}&-0.188008494026&, \\ Q_{\mathbf{4c}}&{}\approx{}&-0.560386158700&{}:{}&1.277245829250&{}:{}&0.283140329449&, \\ I^\prime_{\mathbf{4c}}&{}\approx{}&0.177781512088&{}:{}&1.146089892707&{}:{}&-0.323871404795&, \end{alignedat} \]
\(I_{\mathbf{c}}\)
\(P_{\mathbf{4c}}\)
\(P^-_{\mathbf{4c}}\)
\(P^+_{\mathbf{4c}}\)
\(Q_{\mathbf{4c}}\)
\(I^\prime_{\mathbf{4c}}\)
4c (310)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4c}}&{}\approx{}&3.814084098071&{}:{}&33.769009176847&{}:{}&-36.583093274918&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.055524282344&{}:{}&1.088838851750&{}:{}&-0.144363134093&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.228437733294&{}:{}&0.548250559905&{}:{}&0.223311706801&, \\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.089334773687&{}:{}&0.983135358694&{}:{}&-0.072470132381&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.285721812565&{}:{}&1.078983481238&{}:{}&-0.364705293803&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.070297902075&{}:{}&1.217293481395&{}:{}&-0.287591383470&, \\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.046322677197&{}:{}&0.908394102911&{}:{}&0.045283219892&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.074293902986&{}:{}&0.657780853077&{}:{}&0.267925243937&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.015685886129&{}:{}&1.134766547696&{}:{}&-0.150452433826&, \\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.818544705828&{}:{}&0.181455294172&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.021260915682&{}:{}&0.000000000000&{}:{}&-1.021260915682&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.781723650696&{}:{}&1.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4c}}}{B^\prime_{\mathbf{4c}}}{C^\prime_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4c}}}{B^{\prime\prime\prime}_{\mathbf{4c}}}{C^{\prime\prime\prime}_{\mathbf{4c}}}\)
\(\triangle{A^*_{\mathbf{4c}}}{B^*_{\mathbf{4c}}}{C^*_{\mathbf{4c}}}\)
4c (310)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4c}}}}&{}\approx{}&11.256336392282&\overrightarrow{{A}{I_{\mathbf{c}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4c}}}}&{}\approx{}&0.044419425875&\overrightarrow{{B}{I_{\mathbf{c}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4c}}}}&{}\approx{}&0.182750186635&\overrightarrow{{C}{I_{\mathbf{c}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{c}}&{}\approx{}&1.250000000000&{}:{}&3.000000000000&{}:{}&-3.250000000000&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&3.814084098071&{}:{}&33.769009176847&{}:{}&-36.583093274918&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.055524282344&{}:{}&1.088838851750&{}:{}&-0.144363134093&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.228437733294&{}:{}&0.548250559905&{}:{}&0.223311706801&. \end{alignedat} \]
4c (310)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4c}}&{}\approx{}&0.058686208501&{}:{}&1.016222916292&{}:{}&-0.074909124793&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.089334773687&{}:{}&0.983135358694&{}:{}&-0.072470132381&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.285721812565&{}:{}&1.078983481238&{}:{}&-0.364705293803&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.070297902075&{}:{}&1.217293481395&{}:{}&-0.287591383470&. \end{alignedat} \]
4c (310)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4c}}&{}\approx{}&0.012995544980&{}:{}&0.940138771310&{}:{}&0.046865683709&,\\ A^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.046322677197&{}:{}&0.908394102911&{}:{}&0.045283219892&,\\B^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.074293902986&{}:{}&0.657780853077&{}:{}&0.267925243937&,\\C^{\prime\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.015685886129&{}:{}&1.134766547696&{}:{}&-0.150452433826&. \end{alignedat} \]
4c (310)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4c}}&{}\approx{}&0.101121793394&{}:{}&1.086886700633&{}:{}&-0.188008494026&,\\ A^\prime_{\mathbf{4c}}&{}\approx{}&3.814084098071&{}:{}&33.769009176847&{}:{}&-36.583093274918&,\\B^\prime_{\mathbf{4c}}&{}\approx{}&0.055524282344&{}:{}&1.088838851750&{}:{}&-0.144363134093&,\\C^\prime_{\mathbf{4c}}&{}\approx{}&0.228437733294&{}:{}&0.548250559905&{}:{}&0.223311706801&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.089334773687&{}:{}&0.983135358694&{}:{}&-0.072470132381&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.285721812565&{}:{}&1.078983481238&{}:{}&-0.364705293803&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.070297902075&{}:{}&1.217293481395&{}:{}&-0.287591383470&, \end{alignedat} \]
4c (310)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4c}}&{}\approx{}&-0.560386158700&{}:{}&1.277245829250&{}:{}&0.283140329449&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.818544705828&{}:{}&0.181455294172&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.021260915682&{}:{}&0.000000000000&{}:{}&-1.021260915682&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.781723650696&{}:{}&1.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4c}}&{}\approx{}&0.177781512088&{}:{}&1.146089892707&{}:{}&-0.323871404795&,\\ A^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.089334773687&{}:{}&0.983135358694&{}:{}&-0.072470132381&,\\B^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.285721812565&{}:{}&1.078983481238&{}:{}&-0.364705293803&,\\C^{\prime\prime}_{\mathbf{4c}}&{}\approx{}&0.070297902075&{}:{}&1.217293481395&{}:{}&-0.287591383470&,\\ A^*_{\mathbf{4c}}&{}\approx{}&0.000000000000&{}:{}&0.818544705828&{}:{}&0.181455294172&,\\B^*_{\mathbf{4c}}&{}\approx{}&2.021260915682&{}:{}&0.000000000000&{}:{}&-1.021260915682&,\\C^*_{\mathbf{4c}}&{}\approx{}&-0.781723650696&{}:{}&1.781723650696&{}:{}&0.000000000000&. \end{alignedat} \]
4c (310)