Hocus Pocus Locus

Geometry Level 5

In the rectangular plane, let A,B,C,D,EA, B, C, D, E be points, such that

  • BC\overline{BC} is parallel to the horizontal axis;
  • Point DD is positioned along the locus;
  • AB=CE|\overline{AB}| = |\overline{CE}|;
  • AE=BO=CO=1|\overline{AE}| = |\overline{BO}| = |\overline{CO}| = 1, where point OO is the intersection point of two axes;
  • AEC=BOC\angle AEC = \angle BOC

As shown on the right, for BCO=15\angle BCO = 15^{\circ}, BD\overline{BD} overlaps BO\overline{BO}. The following are all possible arrangements, involving 0θ<300^{\circ} \leq \theta < 30^{\circ} and 30θ<6030^{\circ} \leq \theta < 60^{\circ} respectively, where θ\theta is one of the acute angles of BOC\bigtriangleup BOC.

If the arc length of the locus bounded by 0θ<600^{\circ} \leq \theta < 60^{\circ} is AA, input 105A\lfloor 10^5 A\rfloor as your answer.

Bonus. Find the exact value of AA.

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