Let the vertices of the triangle are and .

Distance between the vertices are side lengths of the triangle.

Calculate the distance between two points by using formula,

.

For :

and

For :

and

For :

and

Conclusion :

Each side of triangle has equal length .

Therefore are vertices of equilateral triangle.

Calculate the mid points of vertices.

Calculate the mid point  by using following formula,

Let mid point of is .

and

Let mid point of is .

and

Let mid point of is .

and

The midpoints are

Calculate the distances between the mid points.

Distance between

Distance between

Distance between

Conclusion :

Each side length of triangle is equal.

are vertices of equilateral triangle.

Equilateral triangle vertices midpoints are also vertices of a second equilateral triangle.