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 > 2b (121)

2b(121)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.002385848374&{}:{}&1.128861538026&{}:{}&-0.126475689652&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.143242893538&{}:{}&0.700263989257&{}:{}&0.156493117205&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.295046141261&{}:{}&1.990185219782&{}:{}&-0.695139078521&, \\ Q_{\mathbf{2b}}&{}\approx{}&-0.508977389101&{}:{}&2.966537620520&{}:{}&-1.457560231419&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.089526731899&{}:{}&1.947384233015&{}:{}&-0.857857501116&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{2b}}\)
\(P^-_{\mathbf{2b}}\)
\(P^+_{\mathbf{2b}}\)
\(Q_{\mathbf{2b}}\)
\(I^\prime_{\mathbf{2b}}\)
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&3.863703305156&{}:{}&3.911891463745&{}:{}&-6.775594768901&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.477013593316&{}:{}&2.303225372841&{}:{}&-0.826211779525&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.926746322053&{}:{}&0.926746322053&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.629837892271&{}:{}&1.835482127752&{}:{}&-0.205644235481&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009229498201&{}:{}&1.498492421539&{}:{}&-0.489262923338&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.005237389824&{}:{}&2.478065243490&{}:{}&-1.472827853667&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.203770941776&{}:{}&0.983893142504&{}:{}&0.219877799271&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.322063909155&{}:{}&0.326080694478&{}:{}&0.351855396368&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.241838298219&{}:{}&1.182262151250&{}:{}&-0.424100449468&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.965925826289&{}:{}&-0.965925826289&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.258819045103&{}:{}&0.000000000000&{}:{}&0.741180954897&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.207106781187&{}:{}&1.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-6.775594768901&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-0.826211779525&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.605171715523&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&3.863703305156&{}:{}&3.911891463745&{}:{}&-6.775594768901&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.477013593316&{}:{}&2.303225372841&{}:{}&-0.826211779525&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.926746322053&{}:{}&0.926746322053&{}:{}&1.000000000000&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.002385848374&{}:{}&1.128861538026&{}:{}&-0.126475689652&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.629837892271&{}:{}&1.835482127752&{}:{}&-0.205644235481&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009229498201&{}:{}&1.498492421539&{}:{}&-0.489262923338&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.005237389824&{}:{}&2.478065243490&{}:{}&-1.472827853667&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.143242893538&{}:{}&0.700263989257&{}:{}&0.156493117205&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.203770941776&{}:{}&0.983893142504&{}:{}&0.219877799271&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.322063909155&{}:{}&0.326080694478&{}:{}&0.351855396368&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.241838298219&{}:{}&1.182262151250&{}:{}&-0.424100449468&. \end{alignedat} \]
2b (121)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.295046141261&{}:{}&1.990185219782&{}:{}&-0.695139078521&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&3.863703305156&{}:{}&3.911891463745&{}:{}&-6.775594768901&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-0.477013593316&{}:{}&2.303225372841&{}:{}&-0.826211779525&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.926746322053&{}:{}&0.926746322053&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.629837892271&{}:{}&1.835482127752&{}:{}&-0.205644235481&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009229498201&{}:{}&1.498492421539&{}:{}&-0.489262923338&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.005237389824&{}:{}&2.478065243490&{}:{}&-1.472827853667&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&-0.508977389101&{}:{}&2.966537620520&{}:{}&-1.457560231419&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.965925826289&{}:{}&-0.965925826289&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.258819045103&{}:{}&0.000000000000&{}:{}&0.741180954897&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.207106781187&{}:{}&1.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.089526731899&{}:{}&1.947384233015&{}:{}&-0.857857501116&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.629837892271&{}:{}&1.835482127752&{}:{}&-0.205644235481&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.009229498201&{}:{}&1.498492421539&{}:{}&-0.489262923338&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.005237389824&{}:{}&2.478065243490&{}:{}&-1.472827853667&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&1.965925826289&{}:{}&-0.965925826289&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.258819045103&{}:{}&0.000000000000&{}:{}&0.741180954897&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.207106781187&{}:{}&1.207106781187&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)