if Z1, Z2 ,Z3 are vertices of equilateral triangle then prove that Z1^2 + Z2^2 + Z3^2 = Z1.Z2 +Z2.Z3+ Z3.Z1.
If Zo is circumcenter of triangle then also prove that 3Zo^2 = Z1^2 + Z2^2 + Z3^2
Answers
Answer:
Since z1, z2, z3, are vertical of an equilateral triangle
therefore circcumcenter (z0) =centroid
(z1+z2+z3) /3...........1)
also for an equilateral triangle
z²1+z²2+z²3=z1z2+z2z3+z3z1........ 2)
on squaring (1). we get,
9z²0=z²1+z²2+z²3+2(z1z2+z2z3+z3z1)
9z²0=z²1+z²2+z²3+2(z²1+z²2+z²3)
3z²0=z²1+z²2+z²3
GIVEN :
TO PROVE :
SOLUTION :
- Let us consider to be the vertices of the equilateral triangle.
We know that, each angle of the equilateral triangle is 60° i.e, .
We have the formula, when the complex number is not rotating about its origin which is,
For, Anticlockwise Direction
As, there the triangle is equilateral, therefore
____________________________________
Now, for Clockwise Direction
Here also,
____________________________________
Now, Multiplying (1) and (2) we get,
Hence Proved .
____________________________________
Now, For Equilateral Triangle,
As is the Circumcentre of the triangle and also the centroid as the triangle is equilateral.
Squaring both sides we get,
(Replacing, by , as we proved earlier we get)