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Symmetry

Math and art by Gábor Damásdi. I usually use the following programs: processing, geogebra, gimp.

We have three colored segment in this animation. Surprisingly the length of the longest one is always the sum of the length of the two smaller ones.
This is actually a very special case of Ptolemy’s theorem. The theorem gives a connection between the... — We have three colored segment in this animation. Surprisingly the length of the longest one is always the sum of the length of the two smaller ones.
This is actually a very special case of Ptolemy’s theorem. The theorem gives a connection between the sides and the diagonals of a cyclic quadrilateral. In this case the length of the dashed lines is equal so the theorem can be simplified to the statement above.

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Morley theorem 2.
The dashed lines are angle trisectors, so they divide the angles of the triangle into three equal parts. The blue triangle is equilateral, which means that each blue segment has the same length.

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Morley’s theorem.
In any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle.
Interestingly I moved the triangle in such a way that the blue triangle doesn’t change its direction.

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I have hidden an Easter egg in this one.
This is based on an other theorem I really like. Given five point on the plane in convex position, one can always draw an ellipse through them. So my algorithm calculates the red points first, and then finds... — I have hidden an Easter egg in this one.
This is based on an other theorem I really like. Given five point on the plane in convex position, one can always draw an ellipse through them. So my algorithm calculates the red points first, and then finds the ellipses through them.
I am currently writing my master thesis on a theorem that is connected to this one.

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Slide.

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Does the inner pentagon have the same size as the others?

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Did you know that you can cut a square into smaller sized squares of different sizes? This is the simplest solution for this problem!

Also note the coloring! By the Four Color Theorem we can color any kind of map with four colors such that neighbours have different colors.

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My collection of animated geometric theorems.

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Seven circle theorem.
Take six circles in a chain that are tangent to a seventh circle and also tangent to their neighbors. Then the three lines drawn between opposite pairs of the points of tangency on the seventh circle all pass through the same... — Seven circle theorem.
Take six circles in a chain that are tangent to a seventh circle and also tangent to their neighbors. Then the three lines drawn between opposite pairs of the points of tangency on the seventh circle all pass through the same point.

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An other nice theorem, try to prove it! Take a triangle (the white one) and build an equilateral triangle on each side of it (the yellow ones). Connect the outer vertices to the opposite vertex in the triangle. Then these three segments meet at the... — An other nice theorem, try to prove it! Take a triangle (the white one) and build an equilateral triangle on each side of it (the yellow ones). Connect the outer vertices to the opposite vertex in the triangle. Then these three segments meet at the same point.

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Take six points on a circle such that every second edge (orange edges) has length equal to the radius of the circle. Then the midpoints of the other three edges form an equilateral triangle.

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