Exploring A Triangle, Its Orthic Triangle, Altitudes and Circumcircle

by Amy Benson

The problem we will examine focuses on the triangle formed by the points of intersection of the extended altitudes of a given triangle ABC and the circumcircle of the same triangle (red triangle). We want to compare this red triangle (labeled altcircumtriangle in our calculations) to the orthic triangle (outlined in blue) of the original triangle ABC.

Focusing solely on the picture of the construction, the two triangles in question appear to be similar. To quickly verify this conjecture, we find the ratio of each pair of corresponding sides. In each instance, the ratio is 0.50, indicating that the triangles are indeed similar. Additionally, we can compare the square of the ratio of the perimeters to the ratio of the areas. We verify that these numbers are indeed the same, another fact that is true of similar figures. It is of further interest to note that the area of the orthic triangle is one fourth the area of the triangle formed by the points of interestion of the extended altitudes and circumcircle. To try this yourself, click here.