If z1,z2,z3∈C are the vertices of an equilateral triangle whose centroid is z0, then k=1∑3(zk−z0)2 is equal to:
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Step-by-Step Solution
Step 1: Centroid Property
The centroid z0 of a triangle with vertices z1,z2,z3 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 2π/3 (or 120 degrees). Therefore, we can express z2−z0 and z3−z0 as rotations of z1−z0 by ei2π/3 and ei4π/3 respectively.
Step 5: Substitute and Simplify
Substitute the rotational relationships into the sum. We use the property that ei8π/3=ei(2π+2π/3)=ei2π/3. The sum of the cube roots of unity (1+ω+ω2) is zero, where ω=ei2π/3. This simplifies the expression to zero.