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 > 1a (013)

1a (013)

Malfatti circles

1a (013)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{1a}}&{}\approx{}&-0.125662529635&{}:{}&0.884079970433&{}:{}&0.241582559203&, \\ P^-_{\mathbf{1a}}&{}\approx{}&-0.043004706904&{}:{}&0.940210803864&{}:{}&0.102793903041&, \\ P^+_{\mathbf{1a}}&{}\approx{}&-0.186173583565&{}:{}&0.842988446646&{}:{}&0.343185136919&, \\ Q_{\mathbf{1a}}&{}\approx{}&0.112819660764&{}:{}&1.116799507471&{}:{}&-0.229619168235&, \\ I^\prime_{\mathbf{1a}}&{}\approx{}&-0.287587527212&{}:{}&0.762803932467&{}:{}&0.524783594745&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{1a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{1a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{1a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{1a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{1a}}\) Radical center of the Malfatti circles
1a (013)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{1a}}&{}\approx{}&-0.175067250635&{}:{}&0.430104464888&{}:{}&0.744962785747&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.103831051254&{}:{}&0.923990395078&{}:{}&0.179840656176&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.027821446333&{}:{}&1.027821446333&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188608749598&{}:{}&0.933517071534&{}:{}&0.255091678064&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.426639284713&{}:{}&0.606437665690&{}:{}&0.820201619023&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.117683902854&{}:{}&0.827947532633&{}:{}&0.289736370221&, \\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.112715286080&{}:{}&1.003051018534&{}:{}&0.109664267546&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.259936737135&{}:{}&0.638611452597&{}:{}&0.621325284538&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.039811660134&{}:{}&0.870401304223&{}:{}&0.169410355910&, \\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.258819045103&{}:{}&-0.258819045103&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.965925826289&{}:{}&0.000000000000&{}:{}&1.965925826289&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{1a}}}{B^\prime_{\mathbf{1a}}}{C^\prime_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{1a}}}{B^{\prime\prime\prime}_{\mathbf{1a}}}{C^{\prime\prime\prime}_{\mathbf{1a}}}\)
\(\triangle{A^*_{\mathbf{1a}}}{B^*_{\mathbf{1a}}}{C^*_{\mathbf{1a}}}\)
1a (013)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{1a}}}}&{}\approx{}&0.744962785747&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{1a}}}}&{}\approx{}&0.179840656176&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{1a}}}}&{}\approx{}&1.780238966158&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.175067250635&{}:{}&0.430104464888&{}:{}&0.744962785747&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.103831051254&{}:{}&0.923990395078&{}:{}&0.179840656176&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.027821446333&{}:{}&1.027821446333&{}:{}&1.000000000000&. \end{alignedat} \]
1a (013)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{1a}}&{}\approx{}&-0.125662529635&{}:{}&0.884079970433&{}:{}&0.241582559203&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188608749598&{}:{}&0.933517071534&{}:{}&0.255091678064&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.426639284713&{}:{}&0.606437665690&{}:{}&0.820201619023&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.117683902854&{}:{}&0.827947532633&{}:{}&0.289736370221&. \end{alignedat} \]
1a (013)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{1a}}&{}\approx{}&-0.043004706904&{}:{}&0.940210803864&{}:{}&0.102793903041&,\\ A^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.112715286080&{}:{}&1.003051018534&{}:{}&0.109664267546&,\\B^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.259936737135&{}:{}&0.638611452597&{}:{}&0.621325284538&,\\C^{\prime\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.039811660134&{}:{}&0.870401304223&{}:{}&0.169410355910&. \end{alignedat} \]
1a (013)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{1a}}&{}\approx{}&-0.186173583565&{}:{}&0.842988446646&{}:{}&0.343185136919&,\\ A^\prime_{\mathbf{1a}}&{}\approx{}&-0.175067250635&{}:{}&0.430104464888&{}:{}&0.744962785747&,\\B^\prime_{\mathbf{1a}}&{}\approx{}&-0.103831051254&{}:{}&0.923990395078&{}:{}&0.179840656176&,\\C^\prime_{\mathbf{1a}}&{}\approx{}&-1.027821446333&{}:{}&1.027821446333&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188608749598&{}:{}&0.933517071534&{}:{}&0.255091678064&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.426639284713&{}:{}&0.606437665690&{}:{}&0.820201619023&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.117683902854&{}:{}&0.827947532633&{}:{}&0.289736370221&, \end{alignedat} \]
1a (013)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{1a}}&{}\approx{}&0.112819660764&{}:{}&1.116799507471&{}:{}&-0.229619168235&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.258819045103&{}:{}&-0.258819045103&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.965925826289&{}:{}&0.000000000000&{}:{}&1.965925826289&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{1a}}&{}\approx{}&-0.287587527212&{}:{}&0.762803932467&{}:{}&0.524783594745&,\\ A^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.188608749598&{}:{}&0.933517071534&{}:{}&0.255091678064&,\\B^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.426639284713&{}:{}&0.606437665690&{}:{}&0.820201619023&,\\C^{\prime\prime}_{\mathbf{1a}}&{}\approx{}&-0.117683902854&{}:{}&0.827947532633&{}:{}&0.289736370221&,\\ A^*_{\mathbf{1a}}&{}\approx{}&0.000000000000&{}:{}&1.258819045103&{}:{}&-0.258819045103&,\\B^*_{\mathbf{1a}}&{}\approx{}&-0.965925826289&{}:{}&0.000000000000&{}:{}&1.965925826289&,\\C^*_{\mathbf{1a}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
1a (013)