Derousseau's Generalization of the Malfatti circles

The Smallest Pythagorean Triangle

\(C=90\degree\).   \(a:b:c=3:4:5\).


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7a(233)

Malfatti circles

7a (233)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&, \\ P_{\mathbf{7a}}&{}\approx{}&1.003192681006&{}:{}&-0.001681803680&{}:{}&-0.001510877327&, \\ P^-_{\mathbf{7a}}&{}\approx{}&0.972508981335&{}:{}&0.011960761241&{}:{}&0.015530257423&, \\ P^+_{\mathbf{7a}}&{}\approx{}&1.035182349490&{}:{}&-0.015905027534&{}:{}&-0.019277321956&, \\ Q_{\mathbf{7a}}&{}\approx{}&0.856523186617&{}:{}&0.085903373829&{}:{}&0.057573439554&, \\ I^\prime_{\mathbf{7a}}&{}\approx{}&1.107885529448&{}:{}&-0.052379304788&{}:{}&-0.055506224660&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{7a}}\)
\(P^-_{\mathbf{7a}}\)
\(P^+_{\mathbf{7a}}\)
\(Q_{\mathbf{7a}}\)
\(I^\prime_{\mathbf{7a}}\)
7a (233)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{7a}}&{}\approx{}&1.032507527261&{}:{}&-0.014447789894&{}:{}&-0.018059737367&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.979228616896&{}:{}&2.319485744597&{}:{}&-3.298714361493&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.841524521636&{}:{}&-1.122032695515&{}:{}&1.280508173879&, \\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.180939510296&{}:{}&-0.095313228485&{}:{}&-0.085626281811&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.030079608459&{}:{}&-0.028528237567&{}:{}&-0.001551370893&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.037662395736&{}:{}&-0.001739590478&{}:{}&-0.035922805258&, \\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.270741005131&{}:{}&0.317285176921&{}:{}&0.411973817947&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.998027544212&{}:{}&-0.013965314428&{}:{}&0.015937770216&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.005219374265&{}:{}&0.012363062102&{}:{}&-0.017582436366&, \\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.598726524541&{}:{}&0.401273475459&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.937016024449&{}:{}&0.000000000000&{}:{}&0.062983975551&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.908848734284&{}:{}&0.091151265716&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{7a}}}{B^\prime_{\mathbf{7a}}}{C^\prime_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{7a}}}{B^{\prime\prime\prime}_{\mathbf{7a}}}{C^{\prime\prime\prime}_{\mathbf{7a}}}\)
\(\triangle{A^*_{\mathbf{7a}}}{B^*_{\mathbf{7a}}}{C^*_{\mathbf{7a}}}\)
7a (233)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{7a}}}}&{}\approx{}&-0.021671684841&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{7a}}}}&{}\approx{}&-3.958457233792&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{7a}}}}&{}\approx{}&-1.683049043273&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.500000000000&{}:{}&0.666666666667&{}:{}&0.833333333333&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.032507527261&{}:{}&-0.014447789894&{}:{}&-0.018059737367&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.979228616896&{}:{}&2.319485744597&{}:{}&-3.298714361493&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.841524521636&{}:{}&-1.122032695515&{}:{}&1.280508173879&. \end{alignedat} \]
7a (233)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{7a}}&{}\approx{}&1.003192681006&{}:{}&-0.001681803680&{}:{}&-0.001510877327&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.180939510296&{}:{}&-0.095313228485&{}:{}&-0.085626281811&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.030079608459&{}:{}&-0.028528237567&{}:{}&-0.001551370893&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.037662395736&{}:{}&-0.001739590478&{}:{}&-0.035922805258&. \end{alignedat} \]
7a (233)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{7a}}&{}\approx{}&0.972508981335&{}:{}&0.011960761241&{}:{}&0.015530257423&,\\ A^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.270741005131&{}:{}&0.317285176921&{}:{}&0.411973817947&,\\B^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&0.998027544212&{}:{}&-0.013965314428&{}:{}&0.015937770216&,\\C^{\prime\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.005219374265&{}:{}&0.012363062102&{}:{}&-0.017582436366&. \end{alignedat} \]
7a (233)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{7a}}&{}\approx{}&1.035182349490&{}:{}&-0.015905027534&{}:{}&-0.019277321956&,\\ A^\prime_{\mathbf{7a}}&{}\approx{}&1.032507527261&{}:{}&-0.014447789894&{}:{}&-0.018059737367&,\\B^\prime_{\mathbf{7a}}&{}\approx{}&1.979228616896&{}:{}&2.319485744597&{}:{}&-3.298714361493&,\\C^\prime_{\mathbf{7a}}&{}\approx{}&0.841524521636&{}:{}&-1.122032695515&{}:{}&1.280508173879&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.180939510296&{}:{}&-0.095313228485&{}:{}&-0.085626281811&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.030079608459&{}:{}&-0.028528237567&{}:{}&-0.001551370893&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.037662395736&{}:{}&-0.001739590478&{}:{}&-0.035922805258&, \end{alignedat} \]
7a (233)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{7a}}&{}\approx{}&0.856523186617&{}:{}&0.085903373829&{}:{}&0.057573439554&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.598726524541&{}:{}&0.401273475459&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.937016024449&{}:{}&0.000000000000&{}:{}&0.062983975551&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.908848734284&{}:{}&0.091151265716&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{7a}}&{}\approx{}&1.107885529448&{}:{}&-0.052379304788&{}:{}&-0.055506224660&,\\ A^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.180939510296&{}:{}&-0.095313228485&{}:{}&-0.085626281811&,\\B^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.030079608459&{}:{}&-0.028528237567&{}:{}&-0.001551370893&,\\C^{\prime\prime}_{\mathbf{7a}}&{}\approx{}&1.037662395736&{}:{}&-0.001739590478&{}:{}&-0.035922805258&,\\ A^*_{\mathbf{7a}}&{}\approx{}&0.000000000000&{}:{}&0.598726524541&{}:{}&0.401273475459&,\\B^*_{\mathbf{7a}}&{}\approx{}&0.937016024449&{}:{}&0.000000000000&{}:{}&0.062983975551&,\\C^*_{\mathbf{7a}}&{}\approx{}&0.908848734284&{}:{}&0.091151265716&{}:{}&0.000000000000&. \end{alignedat} \]
7a (233)