# Equilateral Triangle Complex Numbers Problem

• nighthelios
Then z1*z1 + z2*z2 = 0 + z2*z2 = z1*z2, but z3 doesn't appear anywhere in the equation.In summary, the problem at hand deals with proving an equation involving the vertices of an equilateral triangle in the Argand plane. However, the given equation is impossible to prove as it is false in general. The speaker provides an example to demonstrate this and suggests double-checking the given question.
nighthelios
it's ic a problem on comlex numbers
IF z1 ,z2 ,z3 are vertices of equilateral triangle in argand plane
then
P.T.
z1*z1 + z2*z2 =z1*z2
i hav 1 soln but it's not tat satisfactory

nighthelios said:
it's ic a problem on comlex numbers
IF z1 ,z2 ,z3 are vertices of equilateral triangle in argand plane
then
P.T.
z1*z1 + z2*z2 =z1*z2
i hav 1 soln but it's not tat satisfactory
Why doesn't z3 appear anywhere in the equation you're trying to prove?

AKG said:
Why doesn't z3 appear anywhere in the equation you're trying to prove?

cos we hav to eliminate z3

No, it's because either you wrote the question wrong or the question given to you was stated wrong. What you've asked to prove is impossible to prove, because it's false in general.

One example would be to take the vertices of the equilateral triangle to be the cube roots of 1: 1, $-\frac{1}{2}+i\frac{\sqrt{3}}{2}$, and $-\frac{1}{2}-i\frac{\sqrt{3}}{2}$. If we take z1= 1, z2= $-\frac{1}{2}+i\frac{\sqrt{3}}{2}$, then z12+ z22= $\frac{1}{2}-i\frac{\sqrt{3}}{2}$ which is not z1z2!

Go back and check exactly what it is you are asked to prove.

Or take z1 = 0, z2 = anything else, and z3 any of the two points in the plane that would make z1, z2, z3 an equilateral triangle.

## 1. What is the "Equilateral Triangle Complex Numbers Problem"?

The "Equilateral Triangle Complex Numbers Problem" is a mathematical problem that involves finding the coordinates of the vertices of an equilateral triangle on the complex plane. It is also known as the "Euler's Triangle Problem" or the "Morley's Miracle Problem".

## 2. How do you solve the "Equilateral Triangle Complex Numbers Problem"?

To solve the "Equilateral Triangle Complex Numbers Problem", you can use the formula: z1 = a + bi, z2 = c + di, z3 = e + fi, where a, b, c, d, e, and f are real numbers. By setting z1, z2, and z3 equal to each other, you can find the coordinates of the vertices of the equilateral triangle.

## 3. What is the significance of the "Equilateral Triangle Complex Numbers Problem"?

The "Equilateral Triangle Complex Numbers Problem" has significant applications in mathematics, physics, and engineering. It is used to solve problems related to symmetry, geometry, and complex numbers. It also has connections to other mathematical concepts such as the Golden Ratio and the Pythagorean Theorem.

## 4. Can the "Equilateral Triangle Complex Numbers Problem" be solved using other methods?

Yes, there are other methods for solving the "Equilateral Triangle Complex Numbers Problem", such as using trigonometry or vector algebra. However, the complex numbers method is the most efficient and elegant way to solve this problem.

## 5. Are there any real-life applications of the "Equilateral Triangle Complex Numbers Problem"?

Yes, the "Equilateral Triangle Complex Numbers Problem" has real-life applications in fields such as engineering, architecture, and computer graphics. It is used to design and construct structures with symmetrical patterns, such as bridges, buildings, and video game environments.

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