Proof without words: Geometric series

[Bipolarity]

On this website http://www.albany.edu/~bd445/Eco_466Y/Slides/Infinite_Geometric_Sum,_Proof_Without_Words.pdf there is a "proof without words" of the sum of the infinite geometric series. However, I don't understand what makes the proof valid. In what order were the constructions done etc.? What were the arguments of the proof?

I am referring to the very first picture with the similar triangles.

BiP

 

[tiny-tim]

Hi Bipolarity!

Triangle PQR is similar to triangle TSP.

So the two tans are the same. ​

 

[Bipolarity]

Hi Bipolarity!

Triangle PQR is similar to triangle TSP.

So the two tans are the same. ​

I get that part, but I don't understand how you can construct the line that is equal to the geometric sum.

BiP

 

[tiny-tim]

oh i see!

those diminishing quadrilaterals are similar to each other …

each has two sides the same, and one side r times those two

 

[Bipolarity]

oh i see!

those diminishing quadrilaterals are similar to each other …

each has two sides the same, and one side r times those two

I don't understand how the quadrilaterals were constructed in the first place?

BiP

 

[tiny-tim]

each quadrilateral is stuck onto the previous one, so they're each reduced by r

 

[Bipolarity]

I get it now. Thanks.

BiP