Derousseau's Generalization of the Malfatti circles

Pythagorean Triangle U

\(C=90\degree\).   \(a:b:c=5:12:13\).


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4(200)

Malfatti circles

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Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&, \\ P_{\mathbf{4}}&{}\approx{}&0.999394264747&{}:{}&0.000269489308&{}:{}&0.000336245945&, \\ P^-_{\mathbf{4}}&{}\approx{}&1.015190670300&{}:{}&-0.007313188717&{}:{}&-0.007877481583&, \\ P^+_{\mathbf{4}}&{}\approx{}&0.984175252544&{}:{}&0.007575003782&{}:{}&0.008249743674&, \\ Q_{\mathbf{4}}&{}\approx{}&1.076421539491&{}:{}&-0.032271867348&{}:{}&-0.044149672143&, \\ I^\prime_{\mathbf{4}}&{}\approx{}&0.947852955761&{}:{}&0.024112871576&{}:{}&0.028034172663&, \end{alignedat} \]
\(I\)
\(P_{\mathbf{4}}\)
\(P^-_{\mathbf{4}}\)
\(P^+_{\mathbf{4}}\)
\(Q_{\mathbf{4}}\)
\(I^\prime_{\mathbf{4}}\)
4 (200)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4}}&{}\approx{}&0.984667392808&{}:{}&0.007359651452&{}:{}&0.007972955740&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.777087866053&{}:{}&-1.797516317790&{}:{}&2.020428451737&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.100871453186&{}:{}&2.642091487647&{}:{}&-2.742962940833&, \\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.911067497140&{}:{}&0.039565732015&{}:{}&0.049366770845&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984990184714&{}:{}&0.014678415591&{}:{}&0.000331399696&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.983851468320&{}:{}&0.000265298152&{}:{}&0.015883233527&, \\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.262827771611&{}:{}&0.607958544815&{}:{}&0.654869226796&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.000285438179&{}:{}&0.007476384647&{}:{}&-0.007761822826&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.999107450670&{}:{}&-0.007197329082&{}:{}&0.008089878411&, \\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.422287584928&{}:{}&0.577712415072&,\\B^*_{\mathbf{4}}&{}\approx{}&1.042769422997&{}:{}&0.000000000000&{}:{}&-0.042769422997&,\\C^*_{\mathbf{4}}&{}\approx{}&1.030907319333&{}:{}&-0.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4}}}{B^\prime_{\mathbf{4}}}{C^\prime_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4}}}{B^{\prime\prime}_{\mathbf{4}}}{C^{\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4}}}{B^{\prime\prime\prime}_{\mathbf{4}}}{C^{\prime\prime\prime}_{\mathbf{4}}}\)
\(\triangle{A^*_{\mathbf{4}}}{B^*_{\mathbf{4}}}{C^*_{\mathbf{4}}}\)
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Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4}}}}&{}\approx{}&0.018399128630&\overrightarrow{{A}{I}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4}}}}&{}\approx{}&4.662527196317&\overrightarrow{{B}{I}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4}}}}&{}\approx{}&6.605228719117&\overrightarrow{{C}{I}}. \end{alignedat} \] \[ \begin{alignedat}{4} I&{}\approx{}&0.166666666667&{}:{}&0.400000000000&{}:{}&0.433333333333&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.984667392808&{}:{}&0.007359651452&{}:{}&0.007972955740&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.777087866053&{}:{}&-1.797516317790&{}:{}&2.020428451737&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.100871453186&{}:{}&2.642091487647&{}:{}&-2.742962940833&. \end{alignedat} \]
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First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4}}&{}\approx{}&0.999394264747&{}:{}&0.000269489308&{}:{}&0.000336245945&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.911067497140&{}:{}&0.039565732015&{}:{}&0.049366770845&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984990184714&{}:{}&0.014678415591&{}:{}&0.000331399696&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.983851468320&{}:{}&0.000265298152&{}:{}&0.015883233527&. \end{alignedat} \]
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First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4}}&{}\approx{}&1.015190670300&{}:{}&-0.007313188717&{}:{}&-0.007877481583&,\\ A^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&-0.262827771611&{}:{}&0.607958544815&{}:{}&0.654869226796&,\\B^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&1.000285438179&{}:{}&0.007476384647&{}:{}&-0.007761822826&,\\C^{\prime\prime\prime}_{\mathbf{4}}&{}\approx{}&0.999107450670&{}:{}&-0.007197329082&{}:{}&0.008089878411&. \end{alignedat} \]
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Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4}}&{}\approx{}&0.984175252544&{}:{}&0.007575003782&{}:{}&0.008249743674&,\\ A^\prime_{\mathbf{4}}&{}\approx{}&0.984667392808&{}:{}&0.007359651452&{}:{}&0.007972955740&,\\B^\prime_{\mathbf{4}}&{}\approx{}&0.777087866053&{}:{}&-1.797516317790&{}:{}&2.020428451737&,\\C^\prime_{\mathbf{4}}&{}\approx{}&1.100871453186&{}:{}&2.642091487647&{}:{}&-2.742962940833&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.911067497140&{}:{}&0.039565732015&{}:{}&0.049366770845&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984990184714&{}:{}&0.014678415591&{}:{}&0.000331399696&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.983851468320&{}:{}&0.000265298152&{}:{}&0.015883233527&, \end{alignedat} \]
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Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4}}&{}\approx{}&1.076421539491&{}:{}&-0.032271867348&{}:{}&-0.044149672143&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.422287584928&{}:{}&0.577712415072&,\\B^*_{\mathbf{4}}&{}\approx{}&1.042769422997&{}:{}&0.000000000000&{}:{}&-0.042769422997&,\\C^*_{\mathbf{4}}&{}\approx{}&1.030907319333&{}:{}&-0.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
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Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4}}&{}\approx{}&0.947852955761&{}:{}&0.024112871576&{}:{}&0.028034172663&,\\ A^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.911067497140&{}:{}&0.039565732015&{}:{}&0.049366770845&,\\B^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.984990184714&{}:{}&0.014678415591&{}:{}&0.000331399696&,\\C^{\prime\prime}_{\mathbf{4}}&{}\approx{}&0.983851468320&{}:{}&0.000265298152&{}:{}&0.015883233527&,\\ A^*_{\mathbf{4}}&{}\approx{}&0.000000000000&{}:{}&0.422287584928&{}:{}&0.577712415072&,\\B^*_{\mathbf{4}}&{}\approx{}&1.042769422997&{}:{}&0.000000000000&{}:{}&-0.042769422997&,\\C^*_{\mathbf{4}}&{}\approx{}&1.030907319333&{}:{}&-0.030907319333&{}:{}&0.000000000000&. \end{alignedat} \]
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