Because: AB is the diameter and ACB angle is the right angle.

So: AC vertical BC

Because it's BC‖OD again.

So: OD is perpendicular to AC at point E, angle CBA= angle EOA, △AEO∽△ACB.

Because: AC=OC, OE=OE, angle OEC= angle OEA.

So: all right-angled triangles OEA are equal to right-angled triangles OEC.

So: OD is the middle vertical line of AC.

So: angle AOE= angle COE, AE=EC.

So: angle EOC= angle CBA

And because: △AEO∽△ACB.

So: ao: ab = OE: BC = 1/2.

Because: AE=EC

So: EC= 1/2AC

So: EC:AC=AO:AB

Because: OC=OA.

So: EC:AC=OC:AB.

Because: AB is the diameter

So: ACB angle =90 degrees = OEC angle.

To sum up: because: EC:AC=OC:AB, angle ACB= angle OEC.

So:: △COE∽△ABC

2, because: AD tangent circle O is in A, and AB is the diameter.

So: DA in a is perpendicular to BA.

Because: AB=2, so: AO= 1.

And because: angle BAD=90 degrees tan angle AOD= root number 3.

So: AOD angle =60 degrees.

So: angle ABC=60 degrees.

Because OB=OC, the triangle OCB is a regular triangle.

So: the angle BOC=60 degrees.

So: arc BC=60/360 circumference of circle O.

So: arc BC= 1/3 pie

So: BOC= 1/6 faction of S plate.

Because the triangle BOC is a regular triangle with a side length of 1

So: s triangle BOC= root number 3/4

So: shadow area =S sector BOC-S triangle BOC= 1/6 pie-root number 3/4.

Alas, I'm a junior, and I forgot the theorem. Many symbols can't be typed. Study hard, this question is not difficult.