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If the circle is inside the square, four.

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Q: If you have a circle and a square what is the maximum number of points of intersection that the shapes can share?
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Can 2 circle and 3 lines get 17 intersection points?

Yes. Draw three line segments so that they cross at three points forming a triangle (with each side extending beyond the vertices of the triangle). Draw one circle to enclose the triangle without touching it to intersect the extended sides at a further 6 points, making 9 points of intersection so far. Draw the second circle slightly shifted (relative to the first) so that it also encloses the triangle (without touching it) creating a further 6 intersection points with the three lines and 2 with the first circle; an additional 8 intersection points making 17 in all.


Between how many pairs of points can you measure a circle's diameter?

There are an infinite number of points on the circumference of a circle and an infinite number of points on a semi-circle so the answer to your question is "An infinite number of pairs of points."


What is a great circle in spherical geometry?

The great circle is the intersection of a sphere and any plane passing through its centre. Given two distinct points on the surface of a sphere, those two points and the centre of the sphere define a plane. [If one of the points is at the antipodes of the other, an infinite number of planes are defined.] The great circle is the circle formed when that plane meets the surface of the sphere.