Question

If \( z_1, z_2, z_3 \) are the vertices of an equilateral triangle and \( z \) is its circumcentre, then

• A. \( \left|\frac{z - z_1}{z - z_2}\right| = \left|\frac{z - z_3}{z - z_1}\right| \)
• B. \( |z - z_1| + |z - z_2| + |z - z_3| = 0 \)
• C. \( \frac{z - z_1}{z - z_2} = \frac{z - z_3}{z - z_1} \)
• D. \( \frac{|z - z_1| + |z - z_2|}{|z - z_3|} = 1 \)

Hint:

When dealing with equilateral triangles in complex plane, remember all vertices are equidistant from the circumcenter. Use modulus equality to your advantage.

Answer 1

The Correct Option is A

Step 1: Properties of equilateral triangle in complex plane
In an equilateral triangle, the distances from the circumcenter to each vertex are equal. So: \[ |z - z_1| = |z - z_2| = |z - z_3| \] Step 2: Use symmetry and equal moduli
Take moduli of the expression: \[ \left| \frac{z - z_1}{z - z_2} \right| = \left| \frac{z - z_3}{z - z_1} \right| \] This holds true as all terms are of equal magnitude from the circumcenter.