Let the complex number z1, z2, z3 be...

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1981Moderate

Let the complex number $z_{1}, z_{2}, z_{3}$ be the vertices of an equilateral triangle. Let $z_{0}$ be the circumcentre of the triangle. Then prove that $z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 3 z_{0}^{2}$ .

Visualized Solution (English)

Visualizing the Equilateral Triangle

Let $z_{1}, z_{2}, z_{3}$ be the vertices of an equilateral triangle.
Let $z_{0}$ be the circumcentre (which is also the centroid for an equilateral triangle).

The Equilateral Condition

For an equilateral triangle, the following identity holds:
$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1}$

Defining the Circumcentre .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } z_{0}

Since the triangle is equilateral, the circumcentre $z_{0}$ is the same as the centroid:
$z_{0} = \frac{z _{1} + z _{2} + z _{3}}{3}$

Squaring the Relation

Squaring both sides of the circumcentre equation:
$z_{0}^{2} = (\frac{z _{1} + z _{2} + z _{3}}{3})^{2}$
Which simplifies to:
$9 z_{0}^{2} = (z_{1} + z_{2} + z_{3})^{2}$

Expanding the Square

Expanding the right side using $(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 (ab + b c + c a)$ :
$9 z_{0}^{2} = z_{1}^{2} + z_{2}^{2} + z_{3}^{2} + 2 (z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1})$

The Final Substitution

Substitute $z_{1} z_{2} + z_{2} z_{3} + z_{3} z_{1} = z_{1}^{2} + z_{2}^{2} + z_{3}^{2}$ into the equation:
$9 z_{0}^{2} = (z_{1}^{2} + z_{2}^{2} + z_{3}^{2}) + 2 (z_{1}^{2} + z_{2}^{2} + z_{3}^{2})$
$9 z_{0}^{2} = 3 (z_{1}^{2} + z_{2}^{2} + z_{3}^{2})$

Conclusion and Takeaway

Dividing both sides by $3$ :
$z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 3 z_{0}^{2}$
Key Takeaway: For an equilateral triangle, the sum of squares of vertices is thrice the square of the circumcentre.
Next Challenge: Can you prove this using the rotation theorem ( $e^{iπ /3}$ )?
Next Challenge: What happens if the circumcentre $z_{0}$ is at the origin ( $0$ )?
Next Challenge: Does a similar relation exist for a square?

00:00 / 00:00

Conceptually Similar Problems

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1983Moderate

View in:English Hindi

Prove that the complex numbers $z_{1}, z_{2}$ and the origin form an equilateral triangle only if $z_{1}^{2} + z_{2}^{2} - z_{1} z_{2} = 0$ .

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1986Moderate

View in:English Hindi

Show that the area of the triangle on the Argand diagram formed by the complex numbers $z, i z$ and $z + i z$ is $\frac{1}{2} ∣ z ∣^{2}$ .

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 2001SModerate

View in:English Hindi

The complex numbers $z_{1}, z_{2}$ and $z_{3}$ satisfying $\frac{z _{1} - z _{3}}{z _{2} - z _{3}} = \frac{1 - i 3}{2}$ are the vertices of a triangle which is

(A)

of area zero

(B)

right-angled isosceles

(C)

equilateral

(D)

obtuse-angled isosceles

Answer:C

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1994Moderate

View in:English Hindi

Suppose $Z_{1}, Z_{2}, Z_{3}$ are the vertices of an equilateral triangle inscribed in the circle $∣ Z ∣ = 2$ . If $Z_{1} = 1 + i 3$ then $Z_{2} = \dots, Z_{3} = \dots$

Answer:

- 2, 1 - i 3

MathematicsGeometrical Applications of Complex NumbersJEE Main 2003Moderate

View in:English Hindi

Let $Z_{1}$ and $Z_{2}$ be two roots of the equation $Z^{2} + a Z + b = 0$ , $Z$ being complex. Further, assume that the origin, $Z_{1}$ and $Z_{2}$ form an equilateral triangle. Then

(A)

a^{2} = 4 b

(B)

a^{2} = b

(C)

a^{2} = 2 b

(D)

a^{2} = 3 b

Answer:D

MathematicsGeometrical Applications of Complex NumbersJEE Advanced 1989Moderate

View in:English Hindi

If $a, b, c$ , are the numbers between 0 and 1 such that the points $z_{1} = a + i, z_{2} = 1 + bi$ and $z_{3} = 0$ form an equilateral triangle, then $a = \dots$ and $b = \dots$

Answer:

2 - 3, 2 - 3

MathematicsTrigonometric Ratios and IdentitiesJEE Advanced 1984Moderate

View in:English Hindi

For a triangle $A B C$ it is given that $cos A + cos B + cos C = \frac{3}{2}$ . Prove that the triangle is equilateral.

MathematicsAlgebraic Operations on Complex NumbersJEE Advanced 1995Easy

View in:English Hindi

If $i z^{3} + z^{2} - z + i = 0$ , then show that $∣ z ∣ = 1$ .

MathematicsConditional IdentitiesJEE Advanced 1998Moderate

View in:English Hindi

Prove that a triangle $A B C$ is equilateral if and only if $tan A + tan B + tan C = 33$ .

MathematicsConditional IdentitiesJEE Advanced 2000Easy

View in:English Hindi