Problem 163 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(z_{1}, z_{2}, z_{3}\) are the vertices of an equilateral triangle in the complex plane and \(z_{0}\) is the centroid, then (A) \(\frac{1}{z_{1}-z_{2}}+\frac{1}{z_{2}-z_{3}}+\frac{1}{z_{3}-z_{1}}=0\) (B) \(\left(z_{1}-z_{2}\right)^{2}+\left(z_{2}-z_{3}\right)^{2}+\left(z_{3}-z_{1}\right)^{2}=0\) (C) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=3 z_{0}^{2}\) (D) \(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}=z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}\)

Step-by-Step Solution

Answer

Options (A), (B), and (D) are correct.

Step 1: Understanding the Centroid Property

The centroid of a triangle \(z_1, z_2, z_3\) is given by \((z_1 + z_2 + z_3)/3\). For the given equilateral triangle, \(z_0 = (z_1 + z_2 + z_3)/3\). This means \(z_1 + z_2 + z_3 = 3z_0\).

Step 2: Checking Option (A)

Option (A) states that \(\frac{1}{z_1 - z_2} + \frac{1}{z_2 - z_3} + \frac{1}{z_3 - z_1} = 0\). The property of vertices of an equilateral triangle in the complex plane is such that this equation holds true due to symmetry and rotational transformation properties of the triangle.

Step 3: Checking Option (B)

Option (B) claims \(\left(z_1 - z_2\right)^2 + \left(z_2 - z_3\right)^2 + \left(z_3 - z_1\right)^2 = 0\). This is indeed true because, in an equilateral triangle, the sum of the squares of the sides leads to cancellation due to equal lengths and phase differences of \frac{2\pi}{3}\.

Step 4: Checking Option (C)

For Option (C), \(z_1^2 + z_2^2 + z_3^2 = 3z_0^2\). This statement is incorrect because, although the sum \(z_1 + z_2 + z_3 = 3z_0\), squaring it would not directly relate the squares of individual terms to \z_0^2\.

Step 5: Checking Option (D)

Option (D) \(z_1^2 + z_2^2 + z_3^2 = z_1z_2 + z_2z_3 + z_3z_1\) is a well-known result for equilateral triangles. Since \(z_1 + z_2 + z_3 = 0\) in transformed coordinates (centroid at origin), and all side vectors equally contribute to pairwise products, this result holds true.

Key Concepts

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are of equal length. This property also ensures that all three internal angles are equal, with each angle measuring 60 degrees. An interesting outcome of this symmetry is that any transformation involving rotation or reflection about its center or vertices results in another equilateral triangle.
In the context of the complex plane, where each vertex of the triangle is represented by a complex number (such as \(z_{1}, z_{2}, z_{3}\)), the equilateral triangle maintains its symmetric properties. The vertices, when viewed as points in the complex plane, outline a perfect triangle where relations and operations between these complex numbers adhere to elegant mathematical properties and results depending on their symmetric nature.

Centroid of Triangle

The centroid of a triangle is the point where the three medians intersect. It is often considered the triangle's center of mass if it were made of a uniform material. In coordinate geometry, and specifically within the complex plane, the centroid can be calculated as the average of the vertices' coordinates.
The centroid of any triangle formed by the vertices \(z_1, z_2, \text{and } z_3\) is given by the complex number \(z_0 = \frac{z_1 + z_2 + z_3}{3}\). For equilateral triangles, the centroid has additional symmetry properties, making it equidistant from all three vertices. This symmetry often simplifies problems involving geometric transformations and rotations.

Complex Plane Geometry

Complex plane geometry melds the principles of geometry with complex numbers, allowing for elegant representation and solution of geometric problems. Every point in the complex plane corresponds to a complex number, where the real part specifies the horizontal position and the imaginary part specifies the vertical position.
Operations with complex numbers often simplify geometric transformations. For instance, multiplying a complex number by \(e^{i\theta}\) represents a rotation by the angle \(\theta\), and adding complex numbers translates points in the plane. In the context of an equilateral triangle, these operations reveal intrinsic relationships due to the symmetry of the triangle's sides and angles.

Symmetry in Geometry

Symmetry in geometry refers to the balanced and proportionate similarity found in two halves of an object or figure. Equilateral triangles are prime examples of symmetry due to their equal sides and angles.
In complex plane geometry, symmetry can lead to simplifications, such as when evaluating sums, differences, or products of vertex coordinates. For equilateral triangles, symmetry ensures that rotations by certain amounts (commonly \(\frac{2\pi}{3}\)) result in the same set of vertices, merely rearranged. This rotational symmetry simplifies many geometric proofs and calculations, such as showing that a certain complex expression related to the triangle evaluates to zero.