Corners are cut off from an equilateral triangle to produce a regular hexagon. Are the sides trisected?

Question

The corners are cut off from an equilateral triangle to produce a regular hexagon. Are the sides of the triangle trisected?

In the actual question, the first line was exactly the same. It was then asked to find the ratio of area of the resulting hexagon and the original triangle. In the several solutions of this problem which I found on the internet, they had started with taking the sides of triangle as trisected by this operation and hence the side length of the hexagon would also be equal to one-third of the side length of triangle.

I have seen some variations of this problem where they had explicitly mentioned that the side was trisected and then hexagon was formed.

On stackexchange, there are problems in which they started by trisecting the sides (they mentioned it in the title) and getting a regular polygon.

My question is, if we cut off corners from the equilateral triangle to form regular hexagon, is it going to trisect the sides of triangle or not?

Answer 1

Yes. If we cut off corners to create a regular hexagon, then each angle of the hexagon is $120^\circ$, meaning that each angle of each removed triangle is $60^\circ$, so these triangles are equilateral.

Now all sides of the hexagon are equal. Each triangle you removed shares a side with the hexagon, so all its sides are equal to the side length of the hexagon. Thus the three parts of each side of the original triangle are equal - two of them are sides of removed triangles and the third is a side of the hexagon.

Answer 2

When you trisect the sides, you remove three equilateral triangles and the sides of the hexagon are equal.

Answer 3

We trisect first side and draw parallels to third side trisecting second side. We do the same to the second side drawing parallels from intersection points to first side to trisect the third side... back to first side same way. Nine smaller triangles form between concurrent lines containing a central hexagon.

Whether or not you retain or cut off corner $\frac19$ area small triangles, trisection of sides is ensured by the three sets of parallel lines due to equi-spaced parallels.

Answer 4

In order to handle this question, be it about the size of the segments or the surface, don't look to this question as "cutting the edges of the equilateral triangle", but as "folding the edges of the equilateral triangle towards the centre": it produces exactly the same result (the equidistant hexagon), but the questions about segment length and surface become trivial.