SOLUTION: Prove that if an equilateral triangle is constructed on each side of a given triangle and the third vertex of each of these triangles is joined to the opposite vertex of the origin

Question 405635: Prove that if an equilateral triangle is constructed on each side of a given triangle and the third vertex of each of these triangles is joined to the opposite vertex of the original triangle, then the three segments determined are congruent.
Answer by richard1234(7193) (Show Source): You can put this solution on YOUR website!
Our diagram looks something like this:

(It may appear that these three lines intersect at a point, but that might not necessarily be the case!)

This may seem like an unusual solution, but I believe it is by far the easiest method, and more efficient than trying to brute-force the problem. I will begin by introducing complex numbers into the problem. We can assume that points A, B, ..., F are points on the complex plane.

Suppose that A is the origin of the complex plane, i.e. A = 0, and that B and C are arbitrary complex numbers. It follows that:

and where (The Greek letter "omega" denotes a root of unity, in this case, a 6th root of unity. These equations basically state that F is equal to point C rotated about , and B is the same as D rotated about .) We want to show that CD and BF have equal magnitudes. Since CD = D - C and BF = F - B (this is similar to subtracting vectors), then we want to show that

|D - C| = |F - B| where |z| denotes the magnitude of the complex number z.

Note that and . We can substitute these into our equation to obtain

|D - C| = |(C - D)*omega|. Here, we note that the magnitude of D-C is the same as C-D (i.e. a complex number in the form a+bi has the same magnitude as the number -a-bi). Also, the magnitude of is equal to which is 1. Hence, the magnitude remains unchanged, and we conclude that |D - C| = |F - B| and that CD = BF. By a simple symmetry argument (e.g. assigning B as the origin to show that BF = AE), we conclude that CD = BF = AE, and we are done.

RELATED QUESTIONS
if an equilateral triangle has 15cm on each side what is the height of the... (answered by MathLover1)

The midpoints of each side of an equilateral triangle are joined forming a smaller... (answered by KMST)

An equilateral triangle 8cm on each side is divided into smaller equilateral triangle... (answered by FrankM)

The perimeters of a square and equilateral triangle are equal. if each side of the square (answered by josgarithmetic)

Suppose that a square and an equilateral triangle have the same perimeter. Each side of... (answered by ankor@dixie-net.com)

What is the altitude of an equilateral triangle, if each side of the triangle has a... (answered by mananth)

a rectangle with area 50 sq cm can be divided into 2 same squares. on each side of the... (answered by macston)

which is the length of the third side of an equilateral triangle if two sides each have a (answered by asuar010)

an equilateral triangle is drawn by joining the midpoints of the sides of another... (answered by KMST)