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Intersecting Vectors and the Dot Product [04/24/1998]

Each of the following geometrical theorems can be proved with vectors, using the dot product...

Line through Two Million Points [11/11/1996]

Two million points are randomly scattered in a circle. Will there always be a straight line that passes through the circle and has a million points on each side?

Maximizing the Number of Rectangles [10/26/2000]

How can I cut a rectangular sheet of paper into a maximum number of smaller rectangles of a given size? Is there an algorithm for this?

Maximizing Window Area [02/24/1997]

Maximize the area of a Norman window (rectangular with a semicircle on top) while minimizing the length of the perimeter.

Maximum Angle between Perpendicular Bisectors [06/11/1999]

Which four points on the circumferences of two non-intersecting circles will yield the maximum angle between the two perpendicular bisectors produced by their joins?

Maximum Area Given Enclosing Lengths [11/21/2001]

I am given the lengths of a set of N line segments and I am supposed to calculate the maximum area that can be enclosed using them. Is there an efficient method of finding the solution?

Optimization: Minimum Area [11/07/1997]

How do you fold a piece of paper (rect. with width a and unlimited length) so one corner just reaches the righthand side for minimum area?

Path Less Than 1 + sqrt(3) [03/18/2003]

Is there a way to connect the four vertices of a square (of side length 1) such that the path travelled is less than 1 + sqrt(3)?

Perimeter of a Reuleaux Triangle [04/15/2001]

How can I find the perimeter of a Reuleaux triangle of width h?

Polar Equation of an Ellipse [12/16/1995]

Find the polar equation of the ellipse with eccentricity 3/4...

Real-World Carpentry and Trigonometry [11/19/2002]

I'm trying to come up with a formula to calculate the height of an arc at the midpoint of the chord that defines it knowing only the length of the arc and the length of the chord.

Reuleaux Curve Applications [05/25/2002]

What is the Reuleaux curve used for?

Rotated Conical Sections [07/15/1999]

I am confused by the XY term used in relation to 'straight' conics. Also how does one find the focus, directrix, etc. with such a rotated conic?

Sum of Two Arcs [01/30/2003]

Three points are taken at random on the circumference of a circle. What is the chance that the sum of any two arcs so determined is greater than the third?

Tangent Lines and Odd Degree Polynomials [07/24/1998]

If p(x) is a polynomial of odd degree, determine whether every point in the plane lies on at least one line tangent to the curve y = p(x).

A Triangle in a Circle [05/26/2000]

Suppose you randomly place 2 points on the circumference of a circle. What is the probability that a 3rd point placed randomly on the circle's circumference will form a triangle that will contain the center of the circle?

Triangle in Randomly Colored Plane [10/28/2002]

Prove: Assume that all points in the real plane are colored white or black at random. No matter how the plane is colored (even all white or all black) there is always at least one triangle whose vertices and center of gravity (all 4 points) are of the SAME color.

Voronoi Diagrams [12/12/2000]

On a Voronoi diagram, how do you know which lines and which parts of those lines you need?

Yes, a Graph Can Touch an Asymptote [06/08/2006]

Is it true that the graph of a function can never touch an asymptote?

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