Suppose I have a collection of semicircles along the $x$-axis such that one of the intersection points of each semicircle with the $x$-axis passes through the origin. Would the curves that are intersect all these semicircles perpendicularly be a collection of circles with diameter along the $y$-axis and the circles lie in the upper half plane and they all touch the $x$ axis at the origin? This is just my hunch, is it possible to make it rigorous?
A collection of semicircles
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geometry
1 Answers
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Well, yes. If two circles intersect in one point orthogonally, then they intersect in a second point, also orthogonally. The two intersection points are reflections of each other across the line that connects the centers.