Hocus Pocus Locus

Geometry Level 5

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

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

As shown on the right, for B C O = 1 5 \angle BCO = 15^{\circ} , B D \overline{BD} overlaps B O \overline{BO} . The following are all possible arrangements, involving 0 θ < 3 0 0^{\circ} \leq \theta < 30^{\circ} and 3 0 θ < 6 0 30^{\circ} \leq \theta < 60^{\circ} respectively, where θ \theta is one of the acute angles of B O C \bigtriangleup BOC .

If the arc length of the locus bounded by 0 θ < 6 0 0^{\circ} \leq \theta < 60^{\circ} is A A , input 1 0 5 A \lfloor 10^5 A\rfloor as your answer.

Bonus. Find the exact value of A A .


The answer is 212093.

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