14 distinct ways to arrange three circles in an affine plane, defined by geometric axioms including the uniqueness of lines connecting distinct points and Playfair’s axiom.
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Is the average of two square roots less than, equal to, or greater than the square root of the average?
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In 1937, mathematician
Victor Thébault found that squares constructed on a parallelogram’s sides yield a square when their centers are connected.
( = with vertices lying on a common circle) has diagonals which are perpendicular, then the perpendicular to a side from the point of intersection of the diagonals will bisect the opposite side (AF = FD). cyclic quadrilateral
Does every simple closed curve in the plane contain the vertices of a square?
No one knows
, but the answer to this question is positive if the curve is sufficiently regular.
The black dashed curve goes through all corners of several green squares.
In geometry, the
isogonic center (aka Fermat–Torricelli point) of a triangle, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible.
For years, mathematicians have worked to demonstrate that x 3+ y 3+ z 3 = k, where k is defined as the numbers from 1 to 100. This theory is true in all cases except for an unproven exception: 42.
By 2016 and over a million hours of computation later, researchers of the UK’s Advanced Computing Research Center had its solution for 42.
More intriguing number facts
The three blue points always lie on a straight line. The blue points are the closest points to the moving red point on the lines. In other words the blue points are the projections of the moving red point to the lines.
Summation of Alternating Inverse Powers of Phi…
List of sums of reciprocals.
The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number.