The Pythagorean MRZKM Theorem


As I was preparing the material for the Chennai GeoGebra Workshop I decided that I will show the theorem that I learnt at the 1st Nordic GeoGebra Conference in Iceland.


The green areas in the Pythagorean Theorem figure are all equal.

I started playing with the figure with GeoGebra and I noticed something funny.


The areas of the red triangle and the blue triangle are equal.

I was really excited and on my workshop I showed it to 25 Indian math teachers. None of them had known about this truth before.

I published my finding in a Finnish Math Facebook group and very soon Kalle Leppälä informed me that the triangle where you draw the squares does not have to be right angle. He also explained the way to prove the theorem.


The red and green areas are equal.

Matti Kalevi Sinisalo found that the result will work on any polygon. (Of course I have not tested it with all the polygons.) So if we start with a pentagon ABCDE and draw the squares with side lengths AB, BC, … DE and draw the quadrangles GIKMO and FHJLN. We will find that the areas are the same.


The red and blue areas are equal.

The Pythagorean MRZKM Theorem


M comes from Mikko, that is me. R is my good friend Revathy Parameswaran who invited me to Chennai GeoGebra Workshop, Z is Zekeriya Karadag who told me to use more geometry in my workshop, K and M come from Facebook group Kalle Leppälä and Matti Kalevi Sinisalo.

The files can be found at GeoGebraTube.

5.8.2015 [The blog database broke down in the beginning of August. I had to recreate the post]
23.10.2015 [some editing and adding Z to the name]

22.10.17 [pictures broke, will fix them some time]

25.2.18 [fixed all the picures]