If z1, z2 and z3 are vertices of an equilateral triangle...

If $z_{1}, z_{2}$ and $z_{3}$ are vertices of an equilateral triangle whose orthocentre is, the origin, then (a) $z_{1}+z_{2}+z_{3}=0$ (b) $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|$ (c) $z_{2} z_{3}+z_{y z}_{1}+z_{1} z_{2}=0$ (d) $O z_{1}$ is perpendicular to $z_{2} z_{1}$.

If $z_{1}, z_{2}$ and $z_{3}$ are vertices of an equilateral triangle whose orthocentre is, the origin, then
(a) $z_{1}+z_{2}+z_{3}=0$
(b) $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|$
(c) $z_{2} z_{3}+z_{y z}_{1}+z_{1} z_{2}=0$
(d) $O z_{1}$ is perpendicular to $z_{2} z_{1}$.

Key Concepts

Complex Numbers and the Complex Plane

Complex numbers can be used to represent points in the two?dimensional plane, allowing for an elegant way to encode geometric properties algebraically. When vertices of a geometric figure are given as complex numbers, operations like addition and multiplication can capture translations, rotations, and reflections, making it easier to analyze properties of the figure.

Properties of Equilateral Triangles

An equilateral triangle has all sides equal and all internal angles equal to 60°. This symmetry implies that several important centers of the triangle, such as the centroid, circumcentre, and orthocentre, coincide. This overlapping of centers leads to specific algebraic relationships among the vertices when expressed as complex numbers.

Triangle Centers and Concurrency

In any triangle, various centers like the centroid, circumcentre, and orthocentre have distinct geometric meanings. For equilateral triangles, these centers are the same point. When one of these centers, such as the orthocentre, is at the origin, it imposes symmetry conditions among the vertices, such as their sum being zero and their magnitudes being equal.

Perpendicularity in the Complex Plane

Perpendicularity between lines (or vectors) in the complex plane can be characterized algebraically using the properties of complex multiplication and conjugation. This concept is used to express conditions like one line being perpendicular to another by considering the product of their corresponding complex representations or by relating their directions through rotations by 90°. This is particularly useful when analyzing altitudes or other perpendicular bisectors in triangles.