Angles of Intersecting Chords
Here's a problem based on an obscure theorem about angles of intersecting chords.
Distance Sums
If we place 3 points on a line segment of length 1 and add up all pair-wise distances between them, what’s the biggest sum we can get?
Centers at a Corner
Sketch the area reachable by the midpoint of a line segment of length 1, if its entire length is confined to the first quadrant (x>=0 and y>=0) of a Cartesian plane, and one of its end points must be on the x- or the y- axis.
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.
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.
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?
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.
Line Stitching
Consider all 1 unit long line segments that have an end point on the line y=0 and the other on x=0.
Inscribed Perimeters
If a quadrilateral has 1 vertex on each side of a square of side 1, its smallest possible perimeter is 2sqrt2.
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.