## My thesis part 3.

We arrived to one the main topics of my thesis: Helly’s theorem. For the sake of simplicity I will explain it in 2-dimension. I will try to use very little of the usual formal math language.

Helly’s theorem: We have some convex shapes drawn in the plane and we know that any three of them have a point in common. Then all of them have a point in common.

Ok, but what are convex shapes? Well, here are some:

And these are not convex:

So, a shape is convex if it has no “dents” in it. More precisely a shape is convex if it contains the segment between any two points of the shape.

Helly’s theorem is a classical result in discrete geometry, and many variations and generalizations are known. In my thesis I create a new generalization.

Try it for yourself! Try to draw four convex shapes such that any three of them have a point in common, but not all of them. What happens if we use arbitrary shapes, instead of convex ones?

(If someone is really interested in my thesis, you can read it here: https://www.cs.elte.hu/blobs/diplomamunkak/msc_mat/2017/damasdi_gabor.pdf)

## My thesis Part 1

Finally I have some time to reply. I will write about my thesis in multiple post.

Lets start with a simple problem, which is closely related to my thesis. Everybody should be able to solve it.

Problem: There is a group of people who like to study together. One day they went to the library, but not at the same time. Everyone went there at some point during the day and left the library some time later. We know that for every pair of people there was a moment when booth of them were at the library. Prove that there was a moment during the day when all of them were at the library.

Try to solve it! I will post a solution in a couple of days.

Here is a hint:

## Steiner chain + Apollonius Circles

Steiner chain +

Apollonius Circles

## Johnson’s Theorem. Pick a point in the plane (the red…

Johnson’s Theorem.

Pick a point in the plane (the red one). Draw three circle through it with equal radius (the black circles). Take the three intersection of these circles (the black points). Draw a circle trough the three point (the red circle). Then, the fourth circle have equal size to the first three.

## My thesis Part 1

Finally I have some time to reply. I will write about my thesis in multiple post.

Lets start with a simple problem, which is closely related to my thesis. Everybody should be able to solve it.

Problem: There is a group of people who like to study together. One day they went to the library, but not at the same time. Everyone went there at some point during the day and left the library some time later. We know that for every pair of people there was a moment when booth of them were at the library. Prove that there was a moment during the day when all of them were at the library.

Try to solve it! I will post a solution in a couple of days.

Here is a hint: