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Math problems after the first book of the ninth grade
Solution:

(1)△ABC is an equilateral triangle.

Prove:

∠∠ABC =∠APC = 60,∠BAC=∠CPB=60,

△ ABC is an equilateral triangle.

(2) When the point P is located at the midpoint of AB, the quadrilateral PBOA is a diamond.

Connect OP,

∠∠AOB = 2∠ACB = 120。

P is the midpoint of AB,

∴∠AOP=∠BOP=60

OA = OP = OB,

∴△OAP and△ △OBP are equilateral triangles.

∴OA=AP=OB=PB,

The quadrilateral PBOA is a diamond.