The Intercept theorem

Thales was a great Greek philosopher and mathematician. It is said that he found this theorem when he tried to calculate the height of pyramids in Egypt. His theorem states than in a triangle, if a line is parallel to one side of a triangle then it divides the other two sides proportionally. Now let's not waste any time and get ready for the demonstration !

Euclid is renowned for writing The Elements, in thirteen books he tried to compile all the mathematic definitions, theorems and proof. He also wrote the proof of the intercept theorem, in his 6th book. So let’s get started:

We have a triangle ODC with (AB)//(DC), now watch:

• AADC= AAID+ ADIC
• ABDC=ABIC+ADIC
• AAID+ ADIC= ABIC+ADIC
• AAID= ABIC

Now let’s continue, let’s call h, the height issued from B
AOAB= (OA*h)/2
AODB= (OD*h)/2
AOAB/ AODB= OA/OD
In the same way, k is the height issued from A.
AOAB= (OB*k)/2
AOAC= (OC*k)/2
AOAB/ AOAC= OB/OC
AODB=ADIA+AOAIB=ABIC+AOAIB=AOAC
We can conclude that: OA/OD=OB/OC

And that's it !

Are you saying that we forgot something? Hmm, you're
right, it's not over yet, we didn't talk about AB/DC, thanks
for remainding us.

Let’s continue, we have (AE)//(OC).
If we use the same method as earlier, we will get:
DA/DO=DE/DC
AO/DO=EC/DC
And ABCE is a parallelogram so we have: EC/DC=AB/DC
OA/OD=AB/DC=OB/OC
And that’s all!

Thanks for watching!

A project by Guillaume LARROUTUROU & Ismaïl RAZACK.