# Barney’s triangles

A puzzle from @Barney_MT:

This turns out to be a bit more demanding than I expected. There are spoilers below the line, showing a solution that took rather more time and space than the final polished version does. Spoilers below the line!

When I’ve got isosceles triangles knocking about, I like to add in some circles. Circle theorems, as much as I think they’re a bit of a gimmick at GCSE, are quite useful in geometry puzzles.

Now, how do these circles help?

### How about that top intersection?

The way the circles are set up, there’s an equilateral triangle formed by points A, B and the upper intersection of the circles - let’s call that point E.

Here’s the clever bit: C lies on the circle centred at B. Since chord AE subtends 60 degrees at B, it must subtend 30 degrees at C, meaning that E lies on the line CD!

That makes it easy.

### Easy?!

Now consider line segment BE, which is a chord of the circle centred at A. It subtends 60 degrees there, so it subtends 30 degrees at D, on the circle’s circumference.

And since CDE is a straight line, angle BDC must be 150 degrees.

I’d be curious to learn whether there’s a slicker way of doing that, perhaps one that doesn’t involve circle theorems! Drop me a comment if you can see one.

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

### One comment on “Barney’s triangles”

• ##### Barney Maunder-Taylor

Oh yes, I remember that particular little nightmare now!! Great work Colin, my thanks to you as ever.

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