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 1−r1 = 1+r+r2+r3....
△TPS is cut into multiple similar triangles with a ratio r. ST is 1+r+r2+r3.... For the purposes of this proof, let the point where line QR meets ST be A.
∠PRQ=∠TRA because they're vertical angles. ∠PQR=∠TAR since they are both right angles. Thus △PRQ∼△TRA by AA Similarity. Since △TPS∼△TRA, △PRQ∼△TPS.
Tinkering with our ratios, we get
QRPQ=SPTS
1−r1=11+r+r2+r3...
Q.E.D
Easy Math Editor
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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+r2+...=1−r1−rn... Hmm...
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Thank you for the feedback, I forgot to add that detail!
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It's really amazing,
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Thanks! It's my first post, so I'm on the lines here. :D
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Nice one for the first post. Keep posting!
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Elegant Maths... that's what i call it
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that's fantastic!!!!!!!!!
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Wow. Nice proof!
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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−r2) which can be simplified to get the same answer.
If we extend upto r^3, we get (1+r+r2)/(1−r3) which again can be simplified to get the same result. And so on......
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