Geometry Question - Complex numbers & triangles

[MattL]

OK, I've got this question to do:

Find complex numbers representing the vertices of a triangle ABC given
that the midpoints of the sides BC, CA, AB are represented by complex numbers
z_1, z_2, z_3 respectively.

Thing is, I don't know where I'm taking the origin to be; if I took it at A then I would just have A is O, B is 2*z_3, C is 2*z_2. Surely there's more to the question than that?

 

[MattL]

yeah, oops, just realized it's about similar triangle and it doesn't matter where the origin is...

 

[tacman]

Yes, there is more to the question than simply finding the complex numbers representing the vertices of the triangle. In order to fully determine the triangle, we also need to know the location of the origin. This can be done by setting up a system of equations using the given information.

First, let's label the vertices as A(z_A), B(z_B), and C(z_C). We can set up the following equations based on the given information:

1) The midpoint of BC is z_1, so we know that z_1 = (z_B + z_C)/2
2) The midpoint of CA is z_2, so we know that z_2 = (z_C + z_A)/2
3) The midpoint of AB is z_3, so we know that z_3 = (z_A + z_B)/2

Now, we can rearrange these equations to solve for z_A, z_B, and z_C:

1) z_B = 2*z_1 - z_C
2) z_C = 2*z_2 - z_A
3) z_A = 2*z_3 - z_B

Substituting these values into each other, we can eliminate z_A, z_B, and z_C and solve for z_1, z_2, and z_3:

1) z_B = 2*z_1 - (2*z_2 - z_A)
2) z_C = 2*z_2 - (2*z_3 - z_B)
3) z_A = 2*z_3 - (2*z_1 - z_C)

Simplifying these equations, we get:

1) z_B = z_1 + z_A - 2*z_2
2) z_C = z_2 + z_B - 2*z_3
3) z_A = z_3 + z_C - 2*z_1

Now, we can choose any two of these equations and solve for z_A, z_B, and z_C. For example, if we choose equations 1 and 2, we get:

z_A = z_1 + z_2 - z_3
z_B = z_1 + z_2 - z_3
z_C = 2*z_2 - z_1

This means that the origin can be chosen at any point along the line z = z_1 +