This expression for the golden ratio is quite common, however, before I produced this post – I think it would’ve been very hard to figure out how to derive it from scratch. There aren’t many quirky proofs like this one on the internet – I am quite certain. I hope you liked reading this post! 😀
Step 8: Now connect the points BC and CD together to form another isosceles triangle…
Step 9: The line BC is equal to r… Now label the line BC…
Step 10: Because the line BC and BD are both equal to r, the triangle BCD is an isosceles triangle. This means that the angles ∠BCD and ∠BDC must both be equivalent. Call these angles beta (β).
Step 11: Prove that the angle at point D is equal to 90 degrees.
Thales’ Theorem is as follows:
Because AC is the diameter of the circle you drew, the angle at the point D (α+β) must be equal to 90 degrees. In more specific and general terms, if you have the points A, C and D lying on a circle – and the line AC is in fact the diameter of this circle – then the angle at point D (α+β) must be a right angle.
Proof (which must be derived using the diagram you’ve created):
All angles within a triangle (in 2 space) must add up to 180 degrees.
Mathematically, this means that:
And as a result, Thales’ theorem must be true. The angle α+β is the angle at point D.
The other day I discovered one more way to derive Pythagoras’ equation from scratch, completely by accident. I was deriving Pythagoras’ equation using the usual method, whilst navigating a diagram similar to the one below, but without (B-A) measurements…
*Note (regarding diagram above): x+y = 90 degrees
The usual method goes like this…
The area of the largest square is:
It is also:
Which means that:
Now, when I added the lengths (B-A) to my diagram, which are included in the diagram above, I discovered a new way to derive Pythagoras’ equation…
I did this by focusing on the area C^2. It turns out that:
I was able to say that:
Obviously, I was quite pleased. Have you discovered other ways in which to derive Pythagoras’ equation??
Video on how to come up with Pythagoras’s equation…
I’d also like to talk about a new video I’ve created (posted below). It’s related to a 4 dimensional hypercube and learning how to train your mind to see things from different mathematical perspectives.
Prior to posting up the video above, I did create another similar video. In the video below, you will see me split a prism into 3 equal parts. This video will interest those who’d like to find the volume of a square based pyramid.