## 14 Ways to Arrange Three Circles

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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## Averaging square roots

Is the average of two square roots less than, equal to, or greater than the square root of the average?

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## Thébault’s Theorem

In 1937, mathematician Victor Thébault found that squares constructed on a parallelogram’s sides yield a square when their centers are connected.

## Brahmagupta’s Theorem

If a cyclic quadrilateral ( = 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). ## Toeplitz’ Conjecture

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.

## Isogonic Center

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. ## Life, the Universe, and Maths

For years, mathematicians have worked to demonstrate that x3+y3+z3 = 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 here.

## Wallace-Simson’s Line Theorem

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.   