Parallelogram in the Argand diagram

Question

I want to show that the line connecting two vertices of an equilateral triangle is parallel to another line,connecting vertices of another equilateral triangle.How to show that two lines are parallel in the Argand Plane?

Answer 1

I don't know, Alexsandra. I'm a mathematical economist. Let me take a stab. Let's use the Cartesian plane with real coordinate u (horizontal axis) and imaginary coordinate v (vertical axis). Recall the point (1,1) represents the complex number w = 1 +i. Now let's construct a rhombus (parallelogram with four equal sides). Put the leading vertex of the rhombus at (0,0) and construct it in the positive quadrant. Let's use polar coordinates, so the angle theta at the vertex (0,0) is theta = pi/3. Let's let the length of each of our sides be r = 1. Apparently, we are going to have problems at the next vertices, one upper and one to the right. They are going to be obtuse angles, not acute. Correct me if I'm wrong, however the angle of each is pi/(1.5) in radians or 120 degrees in degrees. So what are we to do? Let's bisect each of these 120 degree angles with a secant starting at the upper vertex to the right of the vertex (0.0) and the lower vertex directly to the right of our initial vertex (0,0). Apparently, we are left with one equilateral triangle with base the secant between (0,0) and (1,0) and another (think of it as inverted with base at the top) equilateral triangle with apex (1,0). Note all four sides of our rhombus have sides with r = 1 and two interior equilateral triangles with interior angles theta = pi/3. By construction, our rhombus is a parallelogram. If you would like to prove it with elementary geometry, try SAS or SSS. I hope this helps.