Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

If you want to upload an image check underneath your post. You will see this input bar. Navigate to where the image is on your hard drive.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,426

hi zoe11221121

To post images you also have to click the **post reply **option not the **quick post** option.

This geometry problem was given to me when I was an A level student many many years ago. My teacher said it is possible to do it by drawing one additional construction line. None of the class could do it and so he showed us. I remember it then looked easy but I cannot remember his method.

Years later I returned to the problem and tried many methods. You can get it by sine and cosine rules, but this would involve using a calculator and so wouldn't give 'absolute accuracy'. I reasoned that, since these rules can be derived from Euclidean geometry, it must be possible to take the rules out of the method and do it using Euclidean geometry itself.

It still took me many years working, on and off, to get there. You will see that there are several isosceles triangles there, and so, many circles can be drawn from a point in the diagram to go through other points in the diagram. One of these creates an equilateral triangle and, for me, that was the breakthrough. You can then find new points on that circle that allow you to complete the problem.

I think the full proof is 'up there' somewhere in this thread if you are still stuck.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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