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 > 4a (211)

4a(211)

Malfatti circles

4a (211)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{4a}}&{}\approx{}&1.128861538026&{}:{}&-0.002385848374&{}:{}&-0.126475689652&, \\ P^-_{\mathbf{4a}}&{}\approx{}&0.700263989257&{}:{}&0.143242893538&{}:{}&0.156493117205&, \\ P^+_{\mathbf{4a}}&{}\approx{}&1.990185219782&{}:{}&-0.295046141261&{}:{}&-0.695139078521&, \\ Q_{\mathbf{4a}}&{}\approx{}&2.966537620520&{}:{}&-0.508977389101&{}:{}&-1.457560231419&, \\ I^\prime_{\mathbf{4a}}&{}\approx{}&1.947384233015&{}:{}&-0.089526731899&{}:{}&-0.857857501116&, \end{alignedat} \]
\(I_{\mathbf{a}}\)
\(P_{\mathbf{4a}}\)
\(P^-_{\mathbf{4a}}\)
\(P^+_{\mathbf{4a}}\)
\(Q_{\mathbf{4a}}\)
\(I^\prime_{\mathbf{4a}}\)
4a (211)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{4a}}&{}\approx{}&2.303225372841&{}:{}&-0.477013593316&{}:{}&-0.826211779525&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&3.911891463745&{}:{}&3.863703305156&{}:{}&-6.775594768901&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.926746322053&{}:{}&-0.926746322053&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.498492421539&{}:{}&-0.009229498201&{}:{}&-0.489262923338&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.835482127752&{}:{}&-0.629837892271&{}:{}&-0.205644235481&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.478065243490&{}:{}&-0.005237389824&{}:{}&-1.472827853667&, \\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.326080694478&{}:{}&0.322063909155&{}:{}&0.351855396368&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.983893142504&{}:{}&-0.203770941776&{}:{}&0.219877799271&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.182262151250&{}:{}&0.241838298219&{}:{}&-0.424100449468&, \\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.258819045103&{}:{}&0.741180954897&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.965925826289&{}:{}&0.000000000000&{}:{}&-0.965925826289&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{4a}}}{B^\prime_{\mathbf{4a}}}{C^\prime_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{4a}}}{B^{\prime\prime\prime}_{\mathbf{4a}}}{C^{\prime\prime\prime}_{\mathbf{4a}}}\)
\(\triangle{A^*_{\mathbf{4a}}}{B^*_{\mathbf{4a}}}{C^*_{\mathbf{4a}}}\)
4a (211)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{4a}}}}&{}\approx{}&-0.826211779525&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{4a}}}}&{}\approx{}&-6.775594768901&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{4a}}}}&{}\approx{}&-1.605171715523&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.303225372841&{}:{}&-0.477013593316&{}:{}&-0.826211779525&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&3.911891463745&{}:{}&3.863703305156&{}:{}&-6.775594768901&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.926746322053&{}:{}&-0.926746322053&{}:{}&1.000000000000&. \end{alignedat} \]
4a (211)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{4a}}&{}\approx{}&1.128861538026&{}:{}&-0.002385848374&{}:{}&-0.126475689652&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.498492421539&{}:{}&-0.009229498201&{}:{}&-0.489262923338&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.835482127752&{}:{}&-0.629837892271&{}:{}&-0.205644235481&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.478065243490&{}:{}&-0.005237389824&{}:{}&-1.472827853667&. \end{alignedat} \]
4a (211)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{4a}}&{}\approx{}&0.700263989257&{}:{}&0.143242893538&{}:{}&0.156493117205&,\\ A^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.326080694478&{}:{}&0.322063909155&{}:{}&0.351855396368&,\\B^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&0.983893142504&{}:{}&-0.203770941776&{}:{}&0.219877799271&,\\C^{\prime\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.182262151250&{}:{}&0.241838298219&{}:{}&-0.424100449468&. \end{alignedat} \]
4a (211)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{4a}}&{}\approx{}&1.990185219782&{}:{}&-0.295046141261&{}:{}&-0.695139078521&,\\ A^\prime_{\mathbf{4a}}&{}\approx{}&2.303225372841&{}:{}&-0.477013593316&{}:{}&-0.826211779525&,\\B^\prime_{\mathbf{4a}}&{}\approx{}&3.911891463745&{}:{}&3.863703305156&{}:{}&-6.775594768901&,\\C^\prime_{\mathbf{4a}}&{}\approx{}&0.926746322053&{}:{}&-0.926746322053&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.498492421539&{}:{}&-0.009229498201&{}:{}&-0.489262923338&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.835482127752&{}:{}&-0.629837892271&{}:{}&-0.205644235481&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.478065243490&{}:{}&-0.005237389824&{}:{}&-1.472827853667&, \end{alignedat} \]
4a (211)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{4a}}&{}\approx{}&2.966537620520&{}:{}&-0.508977389101&{}:{}&-1.457560231419&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.258819045103&{}:{}&0.741180954897&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.965925826289&{}:{}&0.000000000000&{}:{}&-0.965925826289&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{4a}}&{}\approx{}&1.947384233015&{}:{}&-0.089526731899&{}:{}&-0.857857501116&,\\ A^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.498492421539&{}:{}&-0.009229498201&{}:{}&-0.489262923338&,\\B^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&1.835482127752&{}:{}&-0.629837892271&{}:{}&-0.205644235481&,\\C^{\prime\prime}_{\mathbf{4a}}&{}\approx{}&2.478065243490&{}:{}&-0.005237389824&{}:{}&-1.472827853667&,\\ A^*_{\mathbf{4a}}&{}\approx{}&0.000000000000&{}:{}&0.258819045103&{}:{}&0.741180954897&,\\B^*_{\mathbf{4a}}&{}\approx{}&1.965925826289&{}:{}&0.000000000000&{}:{}&-0.965925826289&,\\C^*_{\mathbf{4a}}&{}\approx{}&1.207106781187&{}:{}&-0.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
4a (211)