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 > 5b (303)

5b(303)

Malfatti circles

5b (303)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&, \\ P_{\mathbf{5b}}&{}\approx{}&0.018563675225&{}:{}&-0.002638628253&{}:{}&0.984074953028&, \\ P^-_{\mathbf{5b}}&{}\approx{}&-0.007914090680&{}:{}&0.024593736402&{}:{}&0.983320354278&, \\ P^+_{\mathbf{5b}}&{}\approx{}&0.042749390091&{}:{}&-0.027513619937&{}:{}&0.984764229846&, \\ Q_{\mathbf{5b}}&{}\approx{}&0.252224378167&{}:{}&0.025479836435&{}:{}&0.722295785398&, \\ I^\prime_{\mathbf{5b}}&{}\approx{}&0.097989987026&{}:{}&-0.036943566834&{}:{}&0.938953579808&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{5b}}\)
\(P^-_{\mathbf{5b}}\)
\(P^+_{\mathbf{5b}}\)
\(Q_{\mathbf{5b}}\)
\(I^\prime_{\mathbf{5b}}\)
5b (303)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{5b}}&{}\approx{}&-0.653359874111&{}:{}&-2.258531589634&{}:{}&3.911891463745&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.275403926508&{}:{}&0.247582480175&{}:{}&0.477013593316&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.027821446333&{}:{}&-0.027821446333&{}:{}&1.000000000000&, \\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.050537562140&{}:{}&-0.002552665263&{}:{}&0.952015103123&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.019532503527&{}:{}&-0.054965803114&{}:{}&1.035433299587&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.174460080124&{}:{}&-0.024797637908&{}:{}&0.850337557784&, \\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.026425333700&{}:{}&0.023755832895&{}:{}&0.949818833405&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.008347766595&{}:{}&-0.028856523493&{}:{}&1.037204290087&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.123534050018&{}:{}&0.383892980988&{}:{}&0.739641069030&, \\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.034074173711&{}:{}&0.965925826289&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.258819045103&{}:{}&0.000000000000&{}:{}&0.741180954897&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{5b}}}{B^\prime_{\mathbf{5b}}}{C^\prime_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{5b}}}{B^{\prime\prime\prime}_{\mathbf{5b}}}{C^{\prime\prime\prime}_{\mathbf{5b}}}\)
\(\triangle{A^*_{\mathbf{5b}}}{B^*_{\mathbf{5b}}}{C^*_{\mathbf{5b}}}\)
5b (303)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{5b}}}}&{}\approx{}&3.911891463745&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{5b}}}}&{}\approx{}&0.477013593316&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{5b}}}}&{}\approx{}&0.048188158589&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.577350269190&{}:{}&-0.577350269190&{}:{}&1.000000000000&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&-0.653359874111&{}:{}&-2.258531589634&{}:{}&3.911891463745&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.275403926508&{}:{}&0.247582480175&{}:{}&0.477013593316&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.027821446333&{}:{}&-0.027821446333&{}:{}&1.000000000000&. \end{alignedat} \]
5b (303)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{5b}}&{}\approx{}&0.018563675225&{}:{}&-0.002638628253&{}:{}&0.984074953028&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.050537562140&{}:{}&-0.002552665263&{}:{}&0.952015103123&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.019532503527&{}:{}&-0.054965803114&{}:{}&1.035433299587&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.174460080124&{}:{}&-0.024797637908&{}:{}&0.850337557784&. \end{alignedat} \]
5b (303)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{5b}}&{}\approx{}&-0.007914090680&{}:{}&0.024593736402&{}:{}&0.983320354278&,\\ A^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.026425333700&{}:{}&0.023755832895&{}:{}&0.949818833405&,\\B^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.008347766595&{}:{}&-0.028856523493&{}:{}&1.037204290087&,\\C^{\prime\prime\prime}_{\mathbf{5b}}&{}\approx{}&-0.123534050018&{}:{}&0.383892980988&{}:{}&0.739641069030&. \end{alignedat} \]
5b (303)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{5b}}&{}\approx{}&0.042749390091&{}:{}&-0.027513619937&{}:{}&0.984764229846&,\\ A^\prime_{\mathbf{5b}}&{}\approx{}&-0.653359874111&{}:{}&-2.258531589634&{}:{}&3.911891463745&,\\B^\prime_{\mathbf{5b}}&{}\approx{}&0.275403926508&{}:{}&0.247582480175&{}:{}&0.477013593316&,\\C^\prime_{\mathbf{5b}}&{}\approx{}&0.027821446333&{}:{}&-0.027821446333&{}:{}&1.000000000000&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.050537562140&{}:{}&-0.002552665263&{}:{}&0.952015103123&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.019532503527&{}:{}&-0.054965803114&{}:{}&1.035433299587&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.174460080124&{}:{}&-0.024797637908&{}:{}&0.850337557784&, \end{alignedat} \]
5b (303)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{5b}}&{}\approx{}&0.252224378167&{}:{}&0.025479836435&{}:{}&0.722295785398&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.034074173711&{}:{}&0.965925826289&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.258819045103&{}:{}&0.000000000000&{}:{}&0.741180954897&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{5b}}&{}\approx{}&0.097989987026&{}:{}&-0.036943566834&{}:{}&0.938953579808&,\\ A^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.050537562140&{}:{}&-0.002552665263&{}:{}&0.952015103123&,\\B^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.019532503527&{}:{}&-0.054965803114&{}:{}&1.035433299587&,\\C^{\prime\prime}_{\mathbf{5b}}&{}\approx{}&0.174460080124&{}:{}&-0.024797637908&{}:{}&0.850337557784&,\\ A^*_{\mathbf{5b}}&{}\approx{}&0.000000000000&{}:{}&0.034074173711&{}:{}&0.965925826289&,\\B^*_{\mathbf{5b}}&{}\approx{}&0.258819045103&{}:{}&0.000000000000&{}:{}&0.741180954897&,\\C^*_{\mathbf{5b}}&{}\approx{}&0.908248290464&{}:{}&0.091751709536&{}:{}&0.000000000000&. \end{alignedat} \]
5b (303)