## Circles and Triangles

Regular readers will know that I love a good puzzle. I love all maths problems, but ones which make me think and get me stuck a bit are by far my favourite. The other day Ed Southall (@solvemymaths) shared this little beauty that did just that:

I thought “Circles and a 3 4 5 triangle – what an awesome puzzle”, I reached for a pen an paper and drew out the puzzle.

I was at a bit of a loss to start with. I did some pythag to work some things out:

Eliminated y and did some algebra:

Wrote out what I knew:

And drew a diagram that didn’t help much:

I then added some additional lines to my original diagram:

Which made me see what I needed to do!

I redrew the important bits (using the knowledge that radii meet tangents at 90 degrees and that the line was 3.2 away from c but the center of the large circle was 2.5 away):

Then considered the left bit first:

Used Pythagoras’s theorem:

Then solved for x:

Then briefly git annoyed at myself because I’d already used x for something else.

I did the same with the other side to find the final radius.

A lovely puzzle using mainly Pythagoras’s theorem, circle theorems and algebra so one that is, in theory at least, accessible to GCSE students.

I hope you enjoyed this one as much as I did!

You could probably get AB, CB and DB more efficiently by finding the area of the triangle in two ways (i.e. using 3 and 4 and also using AB and 5)

As for the rest, it seems a bit inelegant. Did anyone come up with a better method?

It does seem a tad inelegant. I’ve not seen anyone else’s method yet but will keep you posted. I feel a coordinate geometry solution may be better I may look at that later.

“Then briefly git annoyed at myself because I’d already used x for something else.”

Indeed, too many variables sometimes creates an impenetrable morass of equations. `Tis often simpler to define variables to reflect the ultimate goal.

Much neater!

Well you’ll probably get a laugh. I tried solving without looking at your post until I was done to see if I came up with something different. (I always find multiple solutions especially when the thinking involved is documented to be interesting.) But on looking through this I did something very similar to above with the same similar triangles and fractional quadratics.