In an equilateral triangle, the vectors from the centroid to the vertices are separated by an angle of 2π/3 (or 120 degrees). Therefore, we can express z_2 - z_0 and z_3 - z_0 as rotations of z_1 - z_0 by e^(i2π/3) and e^(i4π/3) respectively.

In an equilateral triangle, the vectors from the centroid to the vertices are separated by an angle of 2π/3 (or 120 degrees). Therefore, we can express z_2 - z_0 and z_3 - z_0 as rotations of z_1 - z_0 by e^(i2π/3) and e^(i4π/3) respectively.

Substitute the rotational relationships into the sum. We use the property that e^(i8π/3) = e^(i(2π + 2π/3)) = e^(i2π/3). The sum of the cube roots of unity (1 + + ²) is zero, where = e^(i2π/3). This simplifies the expression to zero.

Substitute the rotational relationships into the sum. We use the property that e^(i8π/3) = e^(i(2π + 2π/3)) = e^(i2π/3). The sum of the cube roots of unity (1 + + ²) is zero, where = e^(i2π/3). This simplifies the expression to zero.

If $z_1, z_2, z_3\in\mathbb{C}$ are the vertices of an equilateral triangle whose centroid is $z_0$ , then $\displaystyle\sum_{k=1}^3 (z_k - z_0)^2$ is equal to:

Answer: $0$

Step-by-step solution

Step 1: Centroid Property

The centroid $z_0$ of a triangle with vertices $z_1, z_2, z_3$ is the average of its vertices. This means that the sum of the vectors from the centroid to each vertex is zero.

Step 2: Express sum of vectors from centroid

From the definition of the centroid, we can rearrange the equation to show that the sum of the differences between each vertex and the centroid is zero. This is a fundamental property of the centroid.

Step 3: Equilateral Triangle Property

For an equilateral triangle, the distance from the centroid to each vertex is equal. Let this common distance be $r$ .

Step 4: Rotation Property for Equilateral Triangle

In an equilateral triangle, the vectors from the centroid to the vertices are separated by an angle of $2\pi/3$ (or 120 degrees). Therefore, we can express $z_2 - z_0$ and $z_3 - z_0$ as rotations of $z_1 - z_0$ by $e^{i2\pi/3}$ and $e^{i4\pi/3}$ respectively.

Step 5: Substitute and Simplify

Substitute the rotational relationships into the sum. We use the property that $e^{i8\pi/3} = e^{i(2\pi + 2\pi/3)} = e^{i2\pi/3}$ . The sum of the cube roots of unity ( $1 + \omega + \omega^2$ ) is zero, where $\omega = e^{i2\pi/3}$ . This simplifies the expression to zero.

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If z1,z2,z3∈Cz_1, z_2, z_3\in\mathbb{C}z1​,z2​,z3​∈C are the vertices of an equilateral triangle whose centroid is z0z_0z0​, then ∑k=13(zk−z0)2\displaystyle\sum_{k=1}^3 (z_k - z_0)^2k=1∑3​(zk​−z0​)2 is equal to: