(2)① Establish a rectangular coordinate system with CB as the X-axis and CA as the Y-axis. Let the right-angle sides of the triangle be A, P(x, a-x), then E(0, a-x), O( 1/2a, 1/2a).

PC 2 = x ^ 2 ten(a-x)2 = 2x ^ 2+a ^ 2-2ax

OE 2 =( 1/2A)2X( 1/2A+X-A)2 = X 2X 1/2A 2-AX

So PC= root 2xOE (algebraic method, but also geometric method, connecting OF proves that OF=OE and OF is perpendicular to OE, then EF= root 2xOE, PC= root 2xOE because EF=PC.

② AP 2 ten BP 2 = 2cp 2

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