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Circles S and T have radii 1, and intersect at A and B. The distance between their centers is square root of 2. Let P be a point on major arc AB of circle S, and let PA and PB intersect circle T again at C and D, respectively. Show that CD is a diameter of circle T.

Hint(s):

Add the centers of the circles to the diagram. Also, to prove that  CD  is a diameter, try to use angle relationships. What angles can you compute in the diagram?

 Dec 17, 2019
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Let L be the center of circle S, and let F be the center of circle T.  Since LA = AF = 1 and LF = sqrt(2), triangle LAF is a 45-45-90 triangle.  This means angle ALF is 45 degrees.  By symmetry, angle BLF is also 45 degrees.

 

Then angle ALB = angle ALF + angle LBF = 45 + 45 = 90 degrees, so arc AB is 90 degrees.  Since angle APB subtends arc AB, angle APB is 90/2 = 45 degrees.  Then by Poncelet's Theorem, triangle PCD is a 45-45-90 triangle, so angle PCD is 90 degrees.  Then angle ACD is 90 degrees, which implies that CD is a diameter of circle T.

 Dec 18, 2019

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