Derousseau's Generalization of the Malfatti circles

Isosceles Triangle with 120° Top Angle

\(A=30\degree\), \(B=30\degree\), \(C=120\degree\).


[Top] > Isosceles Triangle with 120° Top Angle > 4b (301)

4b(301)

Malfatti circles

4b (301)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{4b}}&{}\approx{}&0.006514562161&{}:{}&-0.002671022744&{}:{}&0.996156460583&, \\ P^-_{\mathbf{4b}}&{}\approx{}&-0.009304180209&{}:{}&0.013254230062&{}:{}&0.996049950148&, \\ P^+_{\mathbf{4b}}&{}\approx{}&0.021502619901&{}:{}&-0.017759997771&{}:{}&0.996257377870&, \\ Q_{\mathbf{4b}}&{}\approx{}&-0.247814512636&{}:{}&0.042518248463&{}:{}&1.205296264173&, \\ I^\prime_{\mathbf{4b}}&{}\approx{}&0.060118012533&{}:{}&-0.038494686889&{}:{}&0.978376674356&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{4b}}\)
\(P^-_{\mathbf{4b}}\)
\(P^+_{\mathbf{4b}}\)
\(Q_{\mathbf{4b}}\)
\(I^\prime_{\mathbf{4b}}\)
4b (301)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4b}}&{}\approx{}&-3.863703305156&{}:{}&-6.643942271314&{}:{}&11.507645576470&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.280861170610&{}:{}&0.232673012021&{}:{}&0.486465817369&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.016062719530&{}:{}&-0.016062719530&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.030387546170&{}:{}&-0.002606839334&{}:{}&0.972219293164&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.006705430221&{}:{}&-0.032047958349&{}:{}&1.025342528128&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.112762936473&{}:{}&-0.046233708510&{}:{}&0.933470772037&, \\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.015604425176&{}:{}&0.012927129082&{}:{}&0.971468445743&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.009584562210&{}:{}&-0.016481409929&{}:{}&1.026065972139&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.293863234455&{}:{}&0.418621611851&{}:{}&0.875241622604&, \\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.034074173711&{}:{}&0.965925826289&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.258819045103&{}:{}&0.000000000000&{}:{}&1.258819045103&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4b}}}{B^\prime_{\mathbf{4b}}}{C^\prime_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4b}}}{B^{\prime\prime\prime}_{\mathbf{4b}}}{C^{\prime\prime\prime}_{\mathbf{4b}}}\)
\(\triangle{A^*_{\mathbf{4b}}}{B^*_{\mathbf{4b}}}{C^*_{\mathbf{4b}}}\)
4b (301)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4b}}}}&{}\approx{}&11.507645576470&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4b}}}}&{}\approx{}&0.486465817369&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4b}}}}&{}\approx{}&0.027821446333&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-3.863703305156&{}:{}&-6.643942271314&{}:{}&11.507645576470&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.280861170610&{}:{}&0.232673012021&{}:{}&0.486465817369&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.016062719530&{}:{}&-0.016062719530&{}:{}&1.000000000000&. \end{alignedat} \]
4b (301)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4b}}&{}\approx{}&0.006514562161&{}:{}&-0.002671022744&{}:{}&0.996156460583&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.030387546170&{}:{}&-0.002606839334&{}:{}&0.972219293164&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.006705430221&{}:{}&-0.032047958349&{}:{}&1.025342528128&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.112762936473&{}:{}&-0.046233708510&{}:{}&0.933470772037&. \end{alignedat} \]
4b (301)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4b}}&{}\approx{}&-0.009304180209&{}:{}&0.013254230062&{}:{}&0.996049950148&,\\ A^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.015604425176&{}:{}&0.012927129082&{}:{}&0.971468445743&,\\B^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.009584562210&{}:{}&-0.016481409929&{}:{}&1.026065972139&,\\C^{\prime\prime\prime}_{\mathbf{4b}}&{}\approx{}&-0.293863234455&{}:{}&0.418621611851&{}:{}&0.875241622604&. \end{alignedat} \]
4b (301)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4b}}&{}\approx{}&0.021502619901&{}:{}&-0.017759997771&{}:{}&0.996257377870&,\\ A^\prime_{\mathbf{4b}}&{}\approx{}&-3.863703305156&{}:{}&-6.643942271314&{}:{}&11.507645576470&,\\B^\prime_{\mathbf{4b}}&{}\approx{}&0.280861170610&{}:{}&0.232673012021&{}:{}&0.486465817369&,\\C^\prime_{\mathbf{4b}}&{}\approx{}&0.016062719530&{}:{}&-0.016062719530&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.030387546170&{}:{}&-0.002606839334&{}:{}&0.972219293164&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.006705430221&{}:{}&-0.032047958349&{}:{}&1.025342528128&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.112762936473&{}:{}&-0.046233708510&{}:{}&0.933470772037&, \end{alignedat} \]
4b (301)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4b}}&{}\approx{}&-0.247814512636&{}:{}&0.042518248463&{}:{}&1.205296264173&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.034074173711&{}:{}&0.965925826289&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.258819045103&{}:{}&0.000000000000&{}:{}&1.258819045103&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4b}}&{}\approx{}&0.060118012533&{}:{}&-0.038494686889&{}:{}&0.978376674356&,\\ A^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.030387546170&{}:{}&-0.002606839334&{}:{}&0.972219293164&,\\B^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.006705430221&{}:{}&-0.032047958349&{}:{}&1.025342528128&,\\C^{\prime\prime}_{\mathbf{4b}}&{}\approx{}&0.112762936473&{}:{}&-0.046233708510&{}:{}&0.933470772037&,\\ A^*_{\mathbf{4b}}&{}\approx{}&0.000000000000&{}:{}&0.034074173711&{}:{}&0.965925826289&,\\B^*_{\mathbf{4b}}&{}\approx{}&-0.258819045103&{}:{}&0.000000000000&{}:{}&1.258819045103&,\\C^*_{\mathbf{4b}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
4b (301)