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 > 3a (033)

3a(033)

Malfatti circles

3a (033)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{3a}}&{}\approx{}&-0.002638628253&{}:{}&0.018563675225&{}:{}&0.984074953028&, \\ P^-_{\mathbf{3a}}&{}\approx{}&0.024593736402&{}:{}&-0.007914090680&{}:{}&0.983320354278&, \\ P^+_{\mathbf{3a}}&{}\approx{}&-0.027513619937&{}:{}&0.042749390091&{}:{}&0.984764229846&, \\ Q_{\mathbf{3a}}&{}\approx{}&0.025479836435&{}:{}&0.252224378167&{}:{}&0.722295785398&, \\ I^\prime_{\mathbf{3a}}&{}\approx{}&-0.036943566834&{}:{}&0.097989987026&{}:{}&0.938953579808&, \end{alignedat} \]
\(I_{\mathbf{a}}\) Incenter
\(P_{\mathbf{3a}}\) First Ajima-Malfatti point
\(P^-_{\mathbf{3a}}\) First Malfatti-Rabinowitz point
\(P^+_{\mathbf{3a}}\) Gergonne point of the Malfatti triangle
\(Q_{\mathbf{3a}}\) Second Malfatti-Rabinowitz point
\(I^\prime_{\mathbf{3a}}\) Radical center of the Malfatti circles
3a (033)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{3a}}&{}\approx{}&0.247582480175&{}:{}&0.275403926508&{}:{}&0.477013593316&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-2.258531589634&{}:{}&-0.653359874111&{}:{}&3.911891463745&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.027821446333&{}:{}&0.027821446333&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.054965803114&{}:{}&0.019532503527&{}:{}&1.035433299587&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.002552665263&{}:{}&0.050537562140&{}:{}&0.952015103123&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.024797637908&{}:{}&0.174460080124&{}:{}&0.850337557784&, \\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.028856523493&{}:{}&-0.008347766595&{}:{}&1.037204290087&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.023755832895&{}:{}&0.026425333700&{}:{}&0.949818833405&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.383892980988&{}:{}&-0.123534050018&{}:{}&0.739641069030&, \\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.258819045103&{}:{}&0.741180954897&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.034074173711&{}:{}&0.000000000000&{}:{}&0.965925826289&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{3a}}}{B^\prime_{\mathbf{3a}}}{C^\prime_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{3a}}}{B^{\prime\prime\prime}_{\mathbf{3a}}}{C^{\prime\prime\prime}_{\mathbf{3a}}}\)
\(\triangle{A^*_{\mathbf{3a}}}{B^*_{\mathbf{3a}}}{C^*_{\mathbf{3a}}}\)
3a (033)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{3a}}}}&{}\approx{}&0.477013593316&\overrightarrow{{A}{I_{\mathbf{a}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{3a}}}}&{}\approx{}&3.911891463745&\overrightarrow{{B}{I_{\mathbf{a}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{3a}}}}&{}\approx{}&0.048188158589&\overrightarrow{{C}{I_{\mathbf{a}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{a}}&{}\approx{}&-0.577350269190&{}:{}&0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.247582480175&{}:{}&0.275403926508&{}:{}&0.477013593316&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-2.258531589634&{}:{}&-0.653359874111&{}:{}&3.911891463745&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.027821446333&{}:{}&0.027821446333&{}:{}&1.000000000000&. \end{alignedat} \]
3a (033)

First Ajima-Malfatti point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{3a}}&{}\approx{}&-0.002638628253&{}:{}&0.018563675225&{}:{}&0.984074953028&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.054965803114&{}:{}&0.019532503527&{}:{}&1.035433299587&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.002552665263&{}:{}&0.050537562140&{}:{}&0.952015103123&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.024797637908&{}:{}&0.174460080124&{}:{}&0.850337557784&. \end{alignedat} \]
3a (033)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{3a}}&{}\approx{}&0.024593736402&{}:{}&-0.007914090680&{}:{}&0.983320354278&,\\ A^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.028856523493&{}:{}&-0.008347766595&{}:{}&1.037204290087&,\\B^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.023755832895&{}:{}&0.026425333700&{}:{}&0.949818833405&,\\C^{\prime\prime\prime}_{\mathbf{3a}}&{}\approx{}&0.383892980988&{}:{}&-0.123534050018&{}:{}&0.739641069030&. \end{alignedat} \]
3a (033)

Gergonne point of the Malfatti triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{3a}}&{}\approx{}&-0.027513619937&{}:{}&0.042749390091&{}:{}&0.984764229846&,\\ A^\prime_{\mathbf{3a}}&{}\approx{}&0.247582480175&{}:{}&0.275403926508&{}:{}&0.477013593316&,\\B^\prime_{\mathbf{3a}}&{}\approx{}&-2.258531589634&{}:{}&-0.653359874111&{}:{}&3.911891463745&,\\C^\prime_{\mathbf{3a}}&{}\approx{}&-0.027821446333&{}:{}&0.027821446333&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.054965803114&{}:{}&0.019532503527&{}:{}&1.035433299587&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.002552665263&{}:{}&0.050537562140&{}:{}&0.952015103123&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.024797637908&{}:{}&0.174460080124&{}:{}&0.850337557784&, \end{alignedat} \]
3a (033)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{3a}}&{}\approx{}&0.025479836435&{}:{}&0.252224378167&{}:{}&0.722295785398&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.258819045103&{}:{}&0.741180954897&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.034074173711&{}:{}&0.000000000000&{}:{}&0.965925826289&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{3a}}&{}\approx{}&-0.036943566834&{}:{}&0.097989987026&{}:{}&0.938953579808&,\\ A^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.054965803114&{}:{}&0.019532503527&{}:{}&1.035433299587&,\\B^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.002552665263&{}:{}&0.050537562140&{}:{}&0.952015103123&,\\C^{\prime\prime}_{\mathbf{3a}}&{}\approx{}&-0.024797637908&{}:{}&0.174460080124&{}:{}&0.850337557784&,\\ A^*_{\mathbf{3a}}&{}\approx{}&0.000000000000&{}:{}&0.258819045103&{}:{}&0.741180954897&,\\B^*_{\mathbf{3a}}&{}\approx{}&0.034074173711&{}:{}&0.000000000000&{}:{}&0.965925826289&,\\C^*_{\mathbf{3a}}&{}\approx{}&0.091751709536&{}:{}&0.908248290464&{}:{}&0.000000000000&. \end{alignedat} \]
3a (033)