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There's a geometric method of solving that problem with a compass, assuming you have a 2-template of the segment:
Given:
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1) Using a compass create a circle at "A" using as its radius the distance AP:
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2) Now using that same radius, make corresponding circles at points B and P to locate the intersections.
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3) Now using a ruler draw a line through the intersection points of each pair of circles. The intersection of those three lines will be the greater circle's center and the radius follows, as shown.
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Yes, that will work but using the formula will be a lot faster and requires no scaling, no template and no drafting.