Viviani's Theorem states that for an equilateral triangle, the sum of the altitudes from any point in the triangle is equal to the altitude from a vertex of the triangle to the other side.
Let be an equilateral triangle and be a point inside the triangle. We label the altitudes from to each of sides , and , and respectively. Since is equilateral, we can say that . Therefore, , and . Since the area of a triangle is the product of its base and altitude, we also have . However, the area of can also be expressed as . Therefore, , so , which means the sum of the altitudes from any point within the triangle is equal to the altitude from the vertex of a triangle.