Coordinating Geometry on Argand Diagram

Algebra Level 2

On an Argand diagram, a point representing $a+bi$ is called an integral point if $a$ and $b$ are integers. Given that point A and B are both integral points on an Argand diagram. Suppose point C is another point on the Argand diagram such that triangle ABC is equilateral.

Can point C be an integral point?

Cannot be determined Yes No

1 solution

Tom Engelsman
Nov 27, 2020

Quick & dirty example is to let $A = ki, B = -ki$ for $k \in \mathbb{Z}$ . Since $A, B$ are symmetric about the Real axis, this requires the point $C$ to lie on this same axis. However, $C = \sqrt{3}k$ in order to form an equilateral triangle with $A$ and $B$ . This contradicts $C$ being an integral point since $Re(C)$ is irrational.

Coordinating Geometry on Argand Diagram

1 solution

0 pending reports