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Problem 4_7

GeoGebra Construction:

Problem 4_7

Given two circles O1(r1) and O2(r2) with one in the interior of the other, and P a point in the region between the two circles, we construct the two circles passing through P tangent to O1(r1) and O2(r2). (1) Let a half-line through T+ intersect the two circles at A and B. Construct the circle through A, B, P, and let it intersect the line T+P at Q. (2) Construct a circle through P and Q to intersect O1(r1), and let the line joining the intersections intersect T+P at T. (3) Construct a circle with center T and radius the square root of T P ·T Q to intersect O1(r1) at A1 and A2. The two circles each passing through P, Q and one of A1, A2 are tangent to both O1(r1) and O2(r2).


I am very confused by this construction. I made it look as similar to the picture given as possible but the breakdown of 1-3 confuses me and I would like a demonstration of this for support if possible!