Combined Chords

Two circles intersect at P and Q.  Let’s call line AB through P a combined chord if AP and PB are chords of the respective circles.  Let’s define the split product of AB as the length of AP times the length of PB.

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Intersecting Circles

If we draw pairs of circles of the same sizes centered at two distinct points and mark where each pair of circles intersects, we’d trace out a straight line.

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Enclosing the Center

If we randomly select 3 vertices of a regular hexagon to form a triangle, what are the respective probabilities that the center of the hexagon is i) inside, ii) outside, iii) on the edge of the triangle?

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Solid Angles

Let’s consider the 3D version of regular planar angles, called solid angles.  If an angle can be thought of as a portion of a full circle, then a solid angle is a portion of a full sphere.

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Inscribed Perimeters

If a quadrilateral has 1 vertex on each side of a square of side 1, its smallest possible perimeter is 2sqrt2.

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Distances to Vertices

In the square below, points that are closer to the center than one of the vertices is coloured yellow.  The area of the yellow region is 1/2 of the area of the square.

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Area-Bisecting Lines

Every line going through the center of a square bisects its area. Does every line going through the center of an equilateral triangle bisect its area too?

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Chord-bisecting Chords

A circle of radius 5 has chord AB with length 6.  A second chord is drawn here and is bisected by AB.

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