The three vertices of a triangle are represented by the...

The three vertices of a triangle are represented by the complex numbers $0, z_{1}$ and $z_{2}$. If the triangle is equilateral, then
(A) $z_{1}^{2}+z_{2}^{2}+z_{1} z_{2}=0$
(B) $z_{1}^{2}+z_{2}^{2}=z_{1} z_{2}$
(C) $z_{2}^{2}-z_{1}^{2}=z_{1} z_{2}$
(D) $z_{1}^{2}-z_{2}^{2}=z_{1} z_{2}$

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Key Concepts

Complex Numbers

Complex numbers extend the real numbers and are expressed in the form a + bi, where i is the imaginary unit. They allow algebraic operations to be linked with geometric interpretations in the complex plane, making them an essential tool for linking algebra with geometry.

Rotation in the Complex Plane

Multiplication by a complex number of unit modulus, an expression of the form e^(i?), represents a rotation in the complex plane by the angle ?. This property is frequently used to describe geometric transformations such as those encountered in constructing and analyzing equilateral triangles.

Equilateral Triangle Geometry

An equilateral triangle is a triangle in which all three sides have the same length and all three interior angles are 60°. When represented in the complex plane, the vertices of an equilateral triangle can be obtained by rotating one vertex about another by 60° (or ±60°). This rotation relationship leads to key algebraic identities connecting the complex coordinates of the vertices.

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