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 > 2a (031)

2a(031)

Malfatti circles

2a (031)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{2a}}&{}\approx{}&-0.002671022744&{}:{}&0.006514562161&{}:{}&0.996156460583&, \\ P^-_{\mathbf{2a}}&{}\approx{}&0.013254230062&{}:{}&-0.009304180209&{}:{}&0.996049950148&, \\ P^+_{\mathbf{2a}}&{}\approx{}&-0.017759997771&{}:{}&0.021502619901&{}:{}&0.996257377870&, \\ Q_{\mathbf{2a}}&{}\approx{}&0.042518248463&{}:{}&-0.247814512636&{}:{}&1.205296264173&, \\ I^\prime_{\mathbf{2a}}&{}\approx{}&-0.038494686889&{}:{}&0.060118012533&{}:{}&0.978376674356&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{2a}}\)
\(P^-_{\mathbf{2a}}\)
\(P^+_{\mathbf{2a}}\)
\(Q_{\mathbf{2a}}\)
\(I^\prime_{\mathbf{2a}}\)
2a (031)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2a}}&{}\approx{}&0.232673012021&{}:{}&0.280861170610&{}:{}&0.486465817369&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-6.643942271314&{}:{}&-3.863703305156&{}:{}&11.507645576470&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.016062719530&{}:{}&0.016062719530&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.032047958349&{}:{}&0.006705430221&{}:{}&1.025342528128&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.002606839334&{}:{}&0.030387546170&{}:{}&0.972219293164&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.046233708510&{}:{}&0.112762936473&{}:{}&0.933470772037&, \\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.016481409929&{}:{}&-0.009584562210&{}:{}&1.026065972139&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.012927129082&{}:{}&0.015604425176&{}:{}&0.971468445743&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.418621611851&{}:{}&-0.293863234455&{}:{}&0.875241622604&, \\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.258819045103&{}:{}&1.258819045103&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.034074173711&{}:{}&0.000000000000&{}:{}&0.965925826289&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.207106781187&{}:{}&1.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2a}}}{B^\prime_{\mathbf{2a}}}{C^\prime_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2a}}}{B^{\prime\prime\prime}_{\mathbf{2a}}}{C^{\prime\prime\prime}_{\mathbf{2a}}}\)
\(\triangle{A^*_{\mathbf{2a}}}{B^*_{\mathbf{2a}}}{C^*_{\mathbf{2a}}}\)
2a (031)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2a}}}}&{}\approx{}&0.486465817369&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2a}}}}&{}\approx{}&11.507645576470&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2a}}}}&{}\approx{}&0.027821446333&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.232673012021&{}:{}&0.280861170610&{}:{}&0.486465817369&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-6.643942271314&{}:{}&-3.863703305156&{}:{}&11.507645576470&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.016062719530&{}:{}&0.016062719530&{}:{}&1.000000000000&. \end{alignedat} \]
2a (031)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2a}}&{}\approx{}&-0.002671022744&{}:{}&0.006514562161&{}:{}&0.996156460583&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.032047958349&{}:{}&0.006705430221&{}:{}&1.025342528128&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.002606839334&{}:{}&0.030387546170&{}:{}&0.972219293164&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.046233708510&{}:{}&0.112762936473&{}:{}&0.933470772037&. \end{alignedat} \]
2a (031)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2a}}&{}\approx{}&0.013254230062&{}:{}&-0.009304180209&{}:{}&0.996049950148&,\\ A^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.016481409929&{}:{}&-0.009584562210&{}:{}&1.026065972139&,\\B^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.012927129082&{}:{}&0.015604425176&{}:{}&0.971468445743&,\\C^{\prime\prime\prime}_{\mathbf{2a}}&{}\approx{}&0.418621611851&{}:{}&-0.293863234455&{}:{}&0.875241622604&. \end{alignedat} \]
2a (031)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2a}}&{}\approx{}&-0.017759997771&{}:{}&0.021502619901&{}:{}&0.996257377870&,\\ A^\prime_{\mathbf{2a}}&{}\approx{}&0.232673012021&{}:{}&0.280861170610&{}:{}&0.486465817369&,\\B^\prime_{\mathbf{2a}}&{}\approx{}&-6.643942271314&{}:{}&-3.863703305156&{}:{}&11.507645576470&,\\C^\prime_{\mathbf{2a}}&{}\approx{}&-0.016062719530&{}:{}&0.016062719530&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.032047958349&{}:{}&0.006705430221&{}:{}&1.025342528128&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.002606839334&{}:{}&0.030387546170&{}:{}&0.972219293164&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.046233708510&{}:{}&0.112762936473&{}:{}&0.933470772037&, \end{alignedat} \]
2a (031)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2a}}&{}\approx{}&0.042518248463&{}:{}&-0.247814512636&{}:{}&1.205296264173&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.258819045103&{}:{}&1.258819045103&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.034074173711&{}:{}&0.000000000000&{}:{}&0.965925826289&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.207106781187&{}:{}&1.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2a}}&{}\approx{}&-0.038494686889&{}:{}&0.060118012533&{}:{}&0.978376674356&,\\ A^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.032047958349&{}:{}&0.006705430221&{}:{}&1.025342528128&,\\B^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.002606839334&{}:{}&0.030387546170&{}:{}&0.972219293164&,\\C^{\prime\prime}_{\mathbf{2a}}&{}\approx{}&-0.046233708510&{}:{}&0.112762936473&{}:{}&0.933470772037&,\\ A^*_{\mathbf{2a}}&{}\approx{}&0.000000000000&{}:{}&-0.258819045103&{}:{}&1.258819045103&,\\B^*_{\mathbf{2a}}&{}\approx{}&0.034074173711&{}:{}&0.000000000000&{}:{}&0.965925826289&,\\C^*_{\mathbf{2a}}&{}\approx{}&-0.207106781187&{}:{}&1.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
2a (031)