Interesting properties relating areas of two triangles

Abstract

This article explores an interesting geometric relationship between the areas of two triangles, in which each vertex of the smaller triangle is located on distinct sides of the larger triangle. We present and demonstrate three unusual properties, using practical examples with notable points, such as the barycenter, incenter, circumcenter, orthocenter and Gergonne point. The results obtained open paths for future geometric investigations, highlighting classical and modern concepts, such as Euler's theorem applied to pedal triangles, which relates the area of ​​triangles to orthogonal projections of a point on the sides of a triangle.