Prove that equilateral triangles are equal if their heights are equal.

Let an equilateral triangle ABC be given. AB = BA = AC.

Let’s draw the height of the HВ to the base of the speaker. In an equilateral triangle, the height coincides with the bisector and the median, therefore, AH = CH = AC / 2.

Let the side of the triangle be “a” and the height “h”.

Consider a right-angled triangle ABН and express the height through the side of the triangle.

By the Pythagorean theorem, a ^ 2 = (a / 2) ^ 2 + h ^ 2.

h = √ (a ^ 2 – (a / 2) ^ 2 = (a * √3) / 2.

Let two equilateral triangles be given, in which the height h is the same, and the sides are equal to a and b, then:

h = (a * √3) / 2.

h = (in * √3) / 2.

Since h are the same, we equate the equalities

(a * √3) / 2 = (b * √3) / 2.

a = b.

Q.E.D.