Derousseau's Generalization of the Malfatti circles

Half of an Equilateral Triangle

\(A=30\degree\), \(B=60\degree\), \(C=90\degree\).


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\(\mathbf{2b}\) \((121)\)

Malfatti circles

2b (121)

Triangle Centers

Approximately,
\[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&, \\ P_{\mathbf{2b}}&{}\approx{}&-0.025507981647&{}:{}&1.352198071612&{}:{}&-0.326690089965&, \\ P^-_{\mathbf{2b}}&{}\approx{}&0.164995189195&{}:{}&0.716186121978&{}:{}&0.118818688827&, \\ P^+_{\mathbf{2b}}&{}\approx{}&-0.383570727389&{}:{}&2.547622714698&{}:{}&-1.164051987309&, \\ Q_{\mathbf{2b}}&{}\approx{}&-89.189485181446&{}:{}&255.412223499293&{}:{}&-165.222738317847&, \\ I^\prime_{\mathbf{2b}}&{}\approx{}&-0.284007169048&{}:{}&2.721395891160&{}:{}&-1.437388722112&, \end{alignedat} \]
\(I_{\mathbf{b}}\)
\(P_{\mathbf{2b}}\)
\(P^-_{\mathbf{2b}}\)
\(P^+_{\mathbf{2b}}\)
\(Q_{\mathbf{2b}}\)
\(I^\prime_{\mathbf{2b}}\)
2b (121)

Central Triangles

Approximately,
\[ \begin{alignedat}{4} A^\prime_{\mathbf{2b}}&{}\approx{}&1.427921023828&{}:{}&2.766124981280&{}:{}&-3.194046005108&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-1.135027805015&{}:{}&4.405083415044&{}:{}&-2.270055610030&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.726779514551&{}:{}&1.258819045103&{}:{}&0.467960469448&, \\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.886144284064&{}:{}&2.487002255795&{}:{}&-0.600857971732&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.052887391723&{}:{}&1.730235642903&{}:{}&-0.677348251181&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.070253249758&{}:{}&3.724179755262&{}:{}&-2.653926505503&, \\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.283694543421&{}:{}&1.101028646734&{}:{}&0.182665896687&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.273425473874&{}:{}&0.529671474248&{}:{}&0.196903051878&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.322185925341&{}:{}&1.398495856470&{}:{}&-0.720681781812&, \\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.831951230074&{}:{}&-1.831951230074&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.350570754639&{}:{}&0.000000000000&{}:{}&0.649429245361&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.536566092485&{}:{}&1.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
\(\triangle{A}{B}{C}\)
\(\triangle{A^\prime_{\mathbf{2b}}}{B^\prime_{\mathbf{2b}}}{C^\prime_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^{\prime\prime\prime}_{\mathbf{2b}}}{B^{\prime\prime\prime}_{\mathbf{2b}}}{C^{\prime\prime\prime}_{\mathbf{2b}}}\)
\(\triangle{A^*_{\mathbf{2b}}}{B^*_{\mathbf{2b}}}{C^*_{\mathbf{2b}}}\)
2b (121)

Angle bisectors

Approximately,
\[ \begin{alignedat}{2} \overrightarrow{{A}{A^\prime_{\mathbf{2b}}}}&{}\approx{}&-2.024944026382&\overrightarrow{{A}{I_{\mathbf{b}}}},\\\overrightarrow{{B}{B^\prime_{\mathbf{2b}}}}&{}\approx{}&-1.439157588755&\overrightarrow{{B}{I_{\mathbf{b}}}},\\\overrightarrow{{C}{C^\prime_{\mathbf{2b}}}}&{}\approx{}&-0.921519498550&\overrightarrow{{C}{I_{\mathbf{b}}}}. \end{alignedat} \] \[ \begin{alignedat}{4} I_{\mathbf{b}}&{}\approx{}&0.788675134595&{}:{}&-1.366025403784&{}:{}&1.577350269190&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&1.427921023828&{}:{}&2.766124981280&{}:{}&-3.194046005108&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-1.135027805015&{}:{}&4.405083415044&{}:{}&-2.270055610030&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.726779514551&{}:{}&1.258819045103&{}:{}&0.467960469448&. \end{alignedat} \]
2b (121)

First Ajima-Malfatti Point

Approximately,
\[ \begin{alignedat}{4} P_{\mathbf{2b}}&{}\approx{}&-0.025507981647&{}:{}&1.352198071612&{}:{}&-0.326690089965&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.886144284064&{}:{}&2.487002255795&{}:{}&-0.600857971732&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.052887391723&{}:{}&1.730235642903&{}:{}&-0.677348251181&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.070253249758&{}:{}&3.724179755262&{}:{}&-2.653926505503&. \end{alignedat} \]
2b (121)

First Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} P^-_{\mathbf{2b}}&{}\approx{}&0.164995189195&{}:{}&0.716186121978&{}:{}&0.118818688827&,\\ A^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.283694543421&{}:{}&1.101028646734&{}:{}&0.182665896687&,\\B^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.273425473874&{}:{}&0.529671474248&{}:{}&0.196903051878&,\\C^{\prime\prime\prime}_{\mathbf{2b}}&{}\approx{}&0.322185925341&{}:{}&1.398495856470&{}:{}&-0.720681781812&. \end{alignedat} \]
2b (121)

Gergonne Point of the Malfatti Triangle

Approximately,
\[ \begin{alignedat}{4} P^+_{\mathbf{2b}}&{}\approx{}&-0.383570727389&{}:{}&2.547622714698&{}:{}&-1.164051987309&,\\ A^\prime_{\mathbf{2b}}&{}\approx{}&1.427921023828&{}:{}&2.766124981280&{}:{}&-3.194046005108&,\\B^\prime_{\mathbf{2b}}&{}\approx{}&-1.135027805015&{}:{}&4.405083415044&{}:{}&-2.270055610030&,\\C^\prime_{\mathbf{2b}}&{}\approx{}&-0.726779514551&{}:{}&1.258819045103&{}:{}&0.467960469448&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.886144284064&{}:{}&2.487002255795&{}:{}&-0.600857971732&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.052887391723&{}:{}&1.730235642903&{}:{}&-0.677348251181&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.070253249758&{}:{}&3.724179755262&{}:{}&-2.653926505503&, \end{alignedat} \]
2b (121)

Second Malfatti-Rabinowitz point

Approximately,
\[ \begin{alignedat}{4} Q_{\mathbf{2b}}&{}\approx{}&-89.189485181446&{}:{}&255.412223499293&{}:{}&-165.222738317847&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.831951230074&{}:{}&-1.831951230074&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.350570754639&{}:{}&0.000000000000&{}:{}&0.649429245361&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.536566092485&{}:{}&1.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Radical center of the Malfatti circles

Approximately,
\[ \begin{alignedat}{4} I^\prime_{\mathbf{2b}}&{}\approx{}&-0.284007169048&{}:{}&2.721395891160&{}:{}&-1.437388722112&,\\ A^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.886144284064&{}:{}&2.487002255795&{}:{}&-0.600857971732&,\\B^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.052887391723&{}:{}&1.730235642903&{}:{}&-0.677348251181&,\\C^{\prime\prime}_{\mathbf{2b}}&{}\approx{}&-0.070253249758&{}:{}&3.724179755262&{}:{}&-2.653926505503&,\\ A^*_{\mathbf{2b}}&{}\approx{}&0.000000000000&{}:{}&2.831951230074&{}:{}&-1.831951230074&,\\B^*_{\mathbf{2b}}&{}\approx{}&0.350570754639&{}:{}&0.000000000000&{}:{}&0.649429245361&,\\C^*_{\mathbf{2b}}&{}\approx{}&-0.536566092485&{}:{}&1.536566092485&{}:{}&0.000000000000&. \end{alignedat} \]
2b (121)

Hiroyasu Kamo