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]



The Chennai files in Google Drive



Tuesday photos

Wednesday photos

Thursday photos



DATE TIME Group A Group B
13/07/15 9.00-10.30  Inauguration  Inauguration
10.30-12.30PM Dr.KaradagStart up and introducing basics

  • Starting with the basic of GeoGebra: Introducing menu and tools
  • How to use dynamic features: Sliders and Free Objects
  • Introducing function and inverse function
  • Creating linear, quadratic, and sqrt functions
  • Decorating objects
  • Dynamic texts and dynamic colors
Mr. Mikko
1.30-2.30pm Ms.Sangeeta Google apps
2.45 – 4.30pm Dr.KaradagLearning trajectory from geometry to calculus

  • Creating linear functions
  • Calculating the area by using geometric approaches
  • Presenting results by using dynamic texts
  • Creating parabola
  • Using calculus to calculate area
  • Introducing upper sum, lower sum, and Riemann approach
Mr. Mikko
14/0715 9.00-11.00am Dr.KaradagAlgebra and calculus

  • Exploring linear function
    • y=mx
    • y=2x+1 and y=mx+n
  • Exploring parabola
  • Exploring ellipse and hyperbola
  • Translation of functions
    • y= abs(x),   y=abs(x)+k   y = and y= abs(x+k) with slider only
Mr. Mikko
11.15-12.30 Ms.sangeeta Google apps Ms.sangeeta Google apps
1.30-3.30 Dr.KaradagMathematics and Art

  • Creating basic geometric objects by using tools
  • Creating basic geometric objects through Euclidean Perspective
  • Transformational functions
    • Reflecting objects with tools
    • Translating objects with vectors (tools)
    • Rotating objects with tools
    • Rotating objects with matrices
    • Translating objects with matrices
    • Dilating objects with tools
  • Creating an ornament by using tools (sequence and list)
  • Creating dynamic ornament by using slider
3.45-4.15 Mr.MikkoQuestion and answer session Dr.KaradagQuestion and answer session
4.15  valedectory

Chennai GeoGebra workshop

It was fun to have my talk with Google Hangouts. I hope the audience liked it.

1 st Workshop Homework

  1. Create a pentagram on a circle. Calculate the sum of the angles B, …, F. Make a theorem and prove it.
  2. Create a pentagram,where the points are not on a circle. Calculate the sum on angles G, …, K. Make a theorem and prove it.
  3. Make an application with lines that you could use in your lesson.
  4. Make an application with functions that you could use in your lesson.
  5. Go to and find one interesting GeoGebra material. If you dare, comment this blog and tell the name of the material and why you liked it. Don’t use your real name when you comment, use a nick. Enter your real email address, only I can see it.

I will add the links to my examples here some day. If you have questions, please ask by commenting this blog post.

2nd Workshop Homework

Please look at my examples at

  1. Create an app with sliders a, b and c to show what happens in a 2nd degree polynomial when the parameters are changed.
  2. Make an app that uses dynamical text and teaches something about the zeros of a 2nd degree polynomial.
  3. Make an app that plots a function of form (x-a)(x-b)(x-c)… using sliders. Add some dynamical text to it.
  4. Make an app that teaches derivatives.
  5. Make an app that teaches what happens to the value of the average and standard deviation if we add or multiply the original the numbers.
  6. Publish at least one of your own material on GeoGebraTube, use File -> Share.

If you have questions, please ask by commenting this blog post.