If X is the midpoint of the segment connecting the centers of the upper and lower arcs, and JK is perpendicular to XH, then show that H is the midpoint of JK.

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## Solution

We have the PoP condition IH*HJ = KH*HL with us at all times.

Let a = IN = NJ, b = KM = ML, c = NH = HM.

Then IH*HJ = (a+c)(a-c) = a² -c²,

KH* HL = (b-c)*(b+c) = b² – c².

Thus a = b, and KH = b-c = a-c = HJ.