# An Argand Diagram Puzzler

On Twitter, @whitehughes posted a nice complex numbers problem:

I really like this complex number geometry question that I came across this morning. There are so many different ways to tackle it but some are definitely more efficient than others! pic.twitter.com/yrt1n9TqBU

— Susan Whitehouse (@Whitehughes) July 10, 2021

Have a go yourself, if you’d like to; below the line are spoilers.

As Susan says, there are several ways to tackle this. Off the top of my head, I can see:

- An algebraic method, finding the equation of the line BC and a circle of the right radius, then solving
- A vectors method, finding the vector AB, a perpendicular vector, and hence BC
- A trigonometric method, solving triangle ABC

There are probably multiple variations on these; I went directly for manipulating complex numbers.

### My method

Let $z = u-v$, which corresponds to the vector $\vec{BA}$. This works out to be $-2 - (2\sqrt{3}-2)\i$.

The point C is in the first quadrant, so we need to rotate AB by 90º *clockwise* – that corresponds to multiplying $z$ by $-i$. (I think this is the nugget of my solution).

But that won’t give us C – it gives us a complex number corresponding to a multiple of a vector between B and C. To get our final answer, we’ll need to double it, and add on $v$.

So:

- $-2zi = (4\sqrt{3} - 4) + 4\i$
- $v-2zi = (4\sqrt{3} - 1) + 6\i$

And we’re done!

### Or nearly

What do we do when we’ve finished a problem, children?

That’s right! We check our answer, and we look for other solutions.

And naturally, I leave those as an exercise; if you come up with anything interesting, let me know!