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 > 1b (103)

1b(103)

Malfatti circles

1b (103)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{1b}}&{}\approx{}&0.884079970433&{}:{}&-0.125662529635&{}:{}&0.241582559203&, \\ P^-_{\mathbf{1b}}&{}\approx{}&0.940210803864&{}:{}&-0.043004706904&{}:{}&0.102793903041&, \\ P^+_{\mathbf{1b}}&{}\approx{}&0.842988446646&{}:{}&-0.186173583565&{}:{}&0.343185136919&, \\ Q_{\mathbf{1b}}&{}\approx{}&1.116799507471&{}:{}&0.112819660764&{}:{}&-0.229619168235&, \\ I^\prime_{\mathbf{1b}}&{}\approx{}&0.762803932467&{}:{}&-0.287587527212&{}:{}&0.524783594745&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{1b}}\)
\(P^-_{\mathbf{1b}}\)
\(P^+_{\mathbf{1b}}\)
\(Q_{\mathbf{1b}}\)
\(I^\prime_{\mathbf{1b}}\)
1b (103)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1b}}&{}\approx{}&0.923990395078&{}:{}&-0.103831051254&{}:{}&0.179840656176&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.430104464888&{}:{}&-0.175067250635&{}:{}&0.744962785747&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.027821446333&{}:{}&-1.027821446333&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.606437665690&{}:{}&-0.426639284713&{}:{}&0.820201619023&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.933517071534&{}:{}&-0.188608749598&{}:{}&0.255091678064&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.827947532633&{}:{}&-0.117683902854&{}:{}&0.289736370221&, \\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.638611452597&{}:{}&-0.259936737135&{}:{}&0.621325284538&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.003051018534&{}:{}&-0.112715286080&{}:{}&0.109664267546&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.870401304223&{}:{}&-0.039811660134&{}:{}&0.169410355910&, \\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.965925826289&{}:{}&1.965925826289&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.258819045103&{}:{}&0.000000000000&{}:{}&-0.258819045103&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1b}}}{B^\prime_{\mathbf{1b}}}{C^\prime_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1b}}}{B^{\prime\prime\prime}_{\mathbf{1b}}}{C^{\prime\prime\prime}_{\mathbf{1b}}}\)
\(\triangle{A^*_{\mathbf{1b}}}{B^*_{\mathbf{1b}}}{C^*_{\mathbf{1b}}}\)
1b (103)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1b}}}}&{}\approx{}&0.179840656176&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1b}}}}&{}\approx{}&0.744962785747&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1b}}}}&{}\approx{}&1.780238966158&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.923990395078&{}:{}&-0.103831051254&{}:{}&0.179840656176&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.430104464888&{}:{}&-0.175067250635&{}:{}&0.744962785747&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.027821446333&{}:{}&-1.027821446333&{}:{}&1.000000000000&. \end{alignedat} \]
1b (103)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1b}}&{}\approx{}&0.884079970433&{}:{}&-0.125662529635&{}:{}&0.241582559203&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.606437665690&{}:{}&-0.426639284713&{}:{}&0.820201619023&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.933517071534&{}:{}&-0.188608749598&{}:{}&0.255091678064&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.827947532633&{}:{}&-0.117683902854&{}:{}&0.289736370221&. \end{alignedat} \]
1b (103)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1b}}&{}\approx{}&0.940210803864&{}:{}&-0.043004706904&{}:{}&0.102793903041&,\\ A^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.638611452597&{}:{}&-0.259936737135&{}:{}&0.621325284538&,\\B^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&1.003051018534&{}:{}&-0.112715286080&{}:{}&0.109664267546&,\\C^{\prime\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.870401304223&{}:{}&-0.039811660134&{}:{}&0.169410355910&. \end{alignedat} \]
1b (103)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1b}}&{}\approx{}&0.842988446646&{}:{}&-0.186173583565&{}:{}&0.343185136919&,\\ A^\prime_{\mathbf{1b}}&{}\approx{}&0.923990395078&{}:{}&-0.103831051254&{}:{}&0.179840656176&,\\B^\prime_{\mathbf{1b}}&{}\approx{}&0.430104464888&{}:{}&-0.175067250635&{}:{}&0.744962785747&,\\C^\prime_{\mathbf{1b}}&{}\approx{}&1.027821446333&{}:{}&-1.027821446333&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.606437665690&{}:{}&-0.426639284713&{}:{}&0.820201619023&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.933517071534&{}:{}&-0.188608749598&{}:{}&0.255091678064&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.827947532633&{}:{}&-0.117683902854&{}:{}&0.289736370221&, \end{alignedat} \]
1b (103)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1b}}&{}\approx{}&1.116799507471&{}:{}&0.112819660764&{}:{}&-0.229619168235&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.965925826289&{}:{}&1.965925826289&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.258819045103&{}:{}&0.000000000000&{}:{}&-0.258819045103&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1b}}&{}\approx{}&0.762803932467&{}:{}&-0.287587527212&{}:{}&0.524783594745&,\\ A^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.606437665690&{}:{}&-0.426639284713&{}:{}&0.820201619023&,\\B^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.933517071534&{}:{}&-0.188608749598&{}:{}&0.255091678064&,\\C^{\prime\prime}_{\mathbf{1b}}&{}\approx{}&0.827947532633&{}:{}&-0.117683902854&{}:{}&0.289736370221&,\\ A^*_{\mathbf{1b}}&{}\approx{}&0.000000000000&{}:{}&-0.965925826289&{}:{}&1.965925826289&,\\B^*_{\mathbf{1b}}&{}\approx{}&1.258819045103&{}:{}&0.000000000000&{}:{}&-0.258819045103&,\\C^*_{\mathbf{1b}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
1b (103)