## 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. The black dashed curve goes through all corners of several green squares.

## 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. Read more

## Inverse Powers of Phi

Summation of Alternating Inverse Powers of Phi… ## Fibonacci Right Triangle

The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number. ## Original Proof of the Pythagorean Theorem

Clever visual proof by Mike Hirschhorn. ## Euler’s Line

Euler’s line” (red) is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red). Other notable points that lie on the Euler line include the de Longchamps point, the Schiffler point, the Exeter point, and the Gossard perspector.

## Thébault’s theorem

If you place squares on the sides of any parallelogram, their centers will always form a square. 