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! ðŸ˜€

In this post I’ll be demonstrating how you can prove that Thales’ Theorem is true. To follow the steps in this post (11 in total), what you will require is a ruler, pair of compasses and a pencil.

Step 1: Draw a random line on a sheet of paper.

Step 2: Place your compass needle on this line, and form a circle.

Step 3: Add 4 points to yourÂ drawing, as shown below…

Step 4: Name the points A, B, C and D as shown…

Step 5: Connect the points A, B and D together toÂ form an isosceles triangle…

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 anglesÂ âˆ BCD andÂ âˆ BDC must both be equivalent. Call these angles beta (Î²).

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 (Î±+Î²) 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 (Î±+Î²) 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 angleÂ Î±+Î² 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 Â 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:

And since:

I was able to say that:

Obviously, I was quite pleased. Have you discovered other ways in which to derive Pythagoras’ equation??

Related:

Video on how to come up with Pythagoras’s equation…

You can now find out how to derive the 3 main cosine rule formulas through a new document that I’ve created called “Cosine Rule Mastery“.

This document can be downloaded free of charge along with “Sine Rule Mastery” which is another document that explains in detail how to come up with the sine rule formula.

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.

GCSE + A Level Mathematics Proofs, Videos and Tutorials.

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