Hardest geometry problem with an unbelievable solution

HARDEST GEOMETRY PROBLEM WITH UNBELIEVABLE SOLUTION.

This is really great question please read carefully.
Question. Take any equilateral triangle and pick a random point inside the triangle.

Draw from each vertex a line to the random point. Two of the three angles at the random point are known, let’s say angles x and y.

If the three line segments from each vertex to the random point were removed out of the original triangle to form a new triangle, what would the new triangle’s angles be?

Try to figure it out. I was unable to do so but I do have the unbelievable solution if you have any trouble.
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Answer To Hard Geometry Problem With Unbelievably Elegant Solution

As a preliminary, three sides completely determine a triangle (by the side-side-side postulate). So if we have any triangle with those side lengths, we need to solve for those three angles.

The amazing trick is to rotate the triangle 60 degrees from any vertex. Suppose we rotate the triangle 60 degrees counter-clockwise from the left bottom vertex.

If we then connect the random point to the rotated random point, we will end up with the exact triangle we are looking for. We will then solve for those angles.

That’s the idea, let’s work out the steps.

Step 1 is to show the two blue line segments form a 60 degree angle. This is a consequence of the equilateral triangle being rotated 60 degrees.

Next, we connect the endpoints of the two blue line segments. The resulting triangle will be an isosceles triangle of the two blue sides, and the vertex angle is 60 degrees. The remaining angles will be equal as they are opposite the blue sides. So each angle will be (180 – 60)/2 = 60 degrees.

As all three angles are 60 degrees, the triangle is equilateral and the final side has the same length as the blue sides.

From there we then know the shaded triangle is a triangle with blue, yellow, and purple sides.

In the large equilateral triangle, x is the angle between purple and blue sides. In the shaded triangle, one angle is then x less the 60 degree angle from the small equilateral triangle.

Then in the large rotated equilateral triangle, y is the angle between yellow and blue sides. In the shaded triangle, another angle is then y less the 60 degree angle from the small equilateral triangle.

As we know two angles are x – 60 and y – 60, we can then find the last angle since all three angles sum to 180.

180 – (x – 60) – (y – 60)

= 300 – x – y

So that’s the unbelievable solution! The three angles are x – 60, y – 60, and 300 – x – y.
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