by **elim** » Mon Feb 14, 2011 5:35 pm

- [G1] $O, P,Q$ co-linear iff $\angle BAC = 60^{\circ}$ where $AP$ is tangent to $\odot O$ at $A$, $C$ is on $BP$, $PD$ bisects $\angle BPA$
- [G2] The circumcentre of $\triangle O_1O_2O_3$ is on $BC$ iff $\angle BAC = 120^{\circ}$ where $O_1,O_2,O_3$ are circumcentres of $\triangle ABC, \triangle ABD, \triangle ACD$ respectively with $AD$ bisects $\angle BAC$
- [G3] $\angle BAC = 60^{\circ}$ if $O,D,I$ are co-linear where $\triangle ABC$ has distinctive sides, $D$ is the middle point of $BC$, $I$ is the escentre with respect to $\angle A$, $O$ is the circumcentre of $\triangle EFA$ with $BC = BF = CE$

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