Area of a triangle and its heights

Abstract

Certainly, the study of the properties of the triangle and its elements plays a preponderant role in the study of plane geometry, also known as synthetic geometry. From this study, for example, the calculation of the area of other figures, such as
parallelogram and trapezoid, among others, occurs quite naturally, since, in general, we can decompose such a plane figure into
triangles, adding the respective areas to obtain the area of the figure in question. Here, in this work, it is presented and discussed a relationship between the area of a triangle and the area of its corresponding triangle whose sides are the inverse of the heights of the triangle. It is shown that the product of these two areas is a numerical constant, a very interesting and novel result. Particular cases are discussed.