Two moving tangent circles can trace ellipses
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.
The Kepler Triangle, Phi and Pi
A “Kepler triangle” is a right triangle having edge lengths in a geometric progression, in which the common ratio is √ϕ, where ϕ represents the golden ratio.
Well, let’s construct a square with side length √ϕ that inscribes a Kepler triangle, that is, a right triangle with edges 1 : √ϕ : ϕ (or approximately 1 : 1.272 : 1.618), as shown in the picture. Draw then the circumcircle of the Kepler triangle (highlighted in orange in the picture) whose diameter is the hypotenuse of the triangle.
Then, the perimeters of the square (4√ϕ≈5.0884) and the circle (πϕ≈5.083) coincide up to an error less than 0.1%. From this, we can get the approximation coincidence π≈4/√ϕ
Fibonacci Right Triangle
The sum of the squares of consecutive Fibonacci numbers is another Fibonacci number.
Solving An Impossible Packing Problem
Doesn’t fit? Reconstruct!
Equi-extended and isoperimetric non-congruent triangles
The picture below shows the ONLY one pair of triangles with the following properties:
· One triangle is a right triangle and one is isosceles,
· All side lengths of both triangles are rational numbers, and
· The perimeters and areas of both triangles are equal.
Icosahedron with golden ratio cross-sections
3 intersecting golden rectangles (1 : φ) will create the vertices of an icosahedron.
Autokinetic Wireframe Cubes
Curiously enough, the cubes don’t move, only the background color changes…
Here is our tutorial to create an amazing autokinetic animation.
Original Proof of the Pythagorean Theorem
Clever visual proof by Mike Hirschhorn.
Visual Proof (sum of cubes)
The sum of the sequence of the first n cubes equals [n(n+1)/2]² as shown below:
1³+2³+3³+…+n³ = (1+2+3+…+n)² = [n(n+1)/2]²
