Geometry Verify that the points (0, 0), (a, 0), and (a/2, √3a/2) are the vertices of an equilateral triangle. Then show that the midpoints of the three sides are the vertices of a second equilateral triangle (refer to Problem 52).

Data from problem 52

An equilateral triangle is one in which all three sides are of equal length. If two vertices of an equilateral triangle are (0, 4) and (0, 0) find the third vertex. How many of these triangles are possible?