Relationship among some elements of triangles
Relationship among some elements of triangles. The triangle DEF is inscribed in an acute-angled triangle ABC. One can drag each of the vertices of the triangles. For each size of these chosen triangles, the screen shows both the perimeter P of the triangle DEF and the ratio 2S/R, where S and R are the area and the circumradius of the triangle ABC. It’s possible to discover that P>=2S/R and that equality is obtained only when the straight lines AD, BE and FC are altitudes in the triangle ABC (i.e. when DEF is the orthic triangle).