# Complex numbers and geometry

There exist two different complex numbers $c_1$ and $c_2$, that together with $2+2i, 5+i$ form the vertices of two equilateral triangles. Find the product $c_1c_2$.

• You should say what you tried, or at least what methods you are expected to use. – Conifold Sep 6 '14 at 23:44
• How can 4 points form the vertices of an equilateral triangle? – dfg Sep 6 '14 at 23:47
• @dfg no, the two given points and c1 OR c2 can make the triangle. The two triangles share the side with the two given points, but they face in different directions. – Asimov Sep 6 '14 at 23:49
• @Asimov I see, thanks. – dfg Sep 6 '14 at 23:51
• @dfg, yeah, i can understand your confusion – Asimov Sep 6 '14 at 23:51

In the complex plane, this is just a simple geometry problem

So, two points of the triangle are (2,2) and (5,1).

The link below shows how to determine a third point given 2 points and the side lengths Hopefully with it you should be able to figure out the rest. Comment if you get lost so I can advise.

Determine third point of triangle when two points and all sides are known?

• So I got all sides to be sqrt(10). What now? – Aditya More Sep 7 '14 at 0:00
• Actually, you are right, that isnt a great example. So, do you know how a dot product works? Well, we know all the side lengths, and that they are supposed to be $\sqrt{10}$ long, we you need to pick coordinates so that the dot product is | x|| y| cosθ where x and y are the vectors (one from the given point to the other, and one to the unknown point) cos 60 degrees is 1/2 and $\sqrt{10}*\sqrt{10}=10$, so the dot product has to be 5 – Asimov Sep 7 '14 at 0:07
• and the dot product can also be computed as $(x_1*x_2)+(y_1*y_2)$ where $(x_1,y_1)$ is one vector, and $(x_2,y_2)$ is the other. – Asimov Sep 7 '14 at 0:09
• So, what you need to do is find coordinates that make it equal 5,and the two solutions should be your two points (yes, its complicated, but it works) – Asimov Sep 7 '14 at 0:10
• So i need to coordinates that make (x1*x2) + (y1*y2) = 5? Do i just guess and check? – Aditya More Sep 7 '14 at 0:16