I personally love the idea of Proofs Without Words! I like how simple, elegant and creative they can be. So here is another one for all you proof enthusiasts. Link
Expansion
In this proof, we see that \(\frac{1}{1-r}\) = \(1 + r + r^2 + r^3 ...\).
\(\triangle TPS\) is cut into multiple similar triangles with a ratio r. \({ST}\) is \(1 + r + r^2 + r^3 ...\). For the purposes of this proof, let the point where line \({QR}\) meets \(ST\) be \(A\).
\(\angle PRQ = \angle TRA\) because they're vertical angles. \(\angle PQR = \angle TAR\) since they are both right angles. Thus \(\triangle PRQ \sim \triangle TRA\) by \(AA\) Similarity. Since \(\triangle TPS \sim \triangle TRA\), \(\triangle PRQ \sim \triangle TPS\).
Tinkering with our ratios, we get \(\frac{PQ}{QR} = \frac{TS}{SP}\)
\(\frac{1}{1-r} = \frac{1 + r + r^2 + r^3 ...}{1}\)
Q.E.D
Note by
Sherry Sarkar
5 years, 1 month ago
Easy Math Editor
*italics*or_italics_**bold**or__bold__paragraph 1
paragraph 2
[example link](https://brilliant.org)> This is a quote# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"2 \times 32^{34}a_{i-1}\frac{2}{3}\sqrt{2}\sum_{i=1}^3\sin \theta\boxed{123}Comments
Sort by:
Top NewestThis is awesome! Although perhaps it is worth noting that \(|r| < 1\) for this to work. I wonder if I could think up a similar geometric proof for the more general formula: \(1 + r + r^{2} +... = \dfrac{1-r^{n}}{1-r}\)... Hmm...
Log in to reply
Thank you for the feedback, I forgot to add that detail!
Log in to reply
It's really amazing,
Log in to reply
Thanks! It's my first post, so I'm on the lines here. :D
Log in to reply
Nice one for the first post. Keep posting!
Log in to reply
Elegant Maths... that's what i call it
Log in to reply
that's fantastic!!!!!!!!!
Log in to reply
Wow. Nice proof!
Log in to reply
We will get the same answer if we extend the dotted triangle towards the right:
If we extend upto r^2, we get \[(1+r)/(1- r^{2} )\] which can be simplified to get the same answer.
If we extend upto r^3, we get \[(1+r+r^{2})/(1-r^{3})\] which again can be simplified to get the same result. And so on......
Log in to reply