This can be proven by using congruent triangles. Because ACX=YCB are two adjacent angles, one angle is <ACB and the other being an angle that forms an equilateral triangle (which is 60 degrees), it means that ACX=<YCB=<ACB+60.

Question

Now we can see that AC=YC, and CX=CB, these are both sets of sides for an equilateral triangle.

This means that triangles ACX and YCB are congruent triangles, this is because they both have two sides that are the same length, which means they are also the same angle. WE can determine that AX=BY because they are corresponding sides. The same argument can be applied to show that AX=CZ.

This proof will work for all triangles, but needs one small clarification when angle ACB>120 degrees. When the angle ACB=120 degrees, the angles ACX and YCB become 180 degrees each, the triangles ACX and YCB become trivial, this means that they will collapse and become straight lines of equal length. Given this the proof will still work.

When the angle ACB>120 degrees, both triangle ACX and YCB will sit completely outside the triangle ABC. Given this the proof should still hold because ACX=<YCB=(<ACB+60), but now we are working with the outside angles instead.

What i have:
This means that triangles ACX and YCB are congruent triangles, this is because they both have two sides that are the same length, which means they are also the same angle. WE can determine that AX=BY because they are corresponding sides. The same argument can be applied to show that AX=CZ.

This proof will work for all triangles, but needs one small clarification when angle ACB>120 degrees. When the angle ACB=120 degrees, the angles ACX and YCB become 180 degrees each, the triangles ACX and YCB become trivial, this means that they will collapse and become straight lines of equal length. Given this the proof will still work.

When the angle ACB>120 degrees, both triangle ACX and YCB will sit completely outside the triangle ABC. Given this the proof should still hold because ACX=<YCB=(<ACB+60), but now we are working with the outside angles instead.

What i need: i do not understand how to draw this