Assignment 8

By : Olamide Alli

Exploration: Given triangle ABC, Construct the orthocenter H. Let points D, E, and F be the feet of the perpendiculars from A, B, C respectively.

Prove the following:

Given the following we know that the triangle ABC consists of three smaller triangles, BCH, ACH, and ABH. Therefore the area of BCH + area of ACH + area of ABH = area of ABC. Divide both sides by ABC.

Because all of the altitudes connect to the sides of the triangle ABC at right angles, we can express the area of BCH, ACH, and ABH in the equation. We can also express the area of ABC in three separate ways because there are three different altitudes for each side.

We have solved our first equation.

Now let’s prove the second equation.

To prove the 2nd equation, by looking at the drawing we know that we can express segments as the difference of two other segments. Therefore HD = AD – AH, HF = BE – BH, and HF = CF – CH. Now let’s substitute these difference equations into

We have proven the second equation.

What if the triangle ABC was obtuse? Will the two proven equations still apply?

Drag point C in the triangle in order to make CAB obtuse, the orthocenter it moves outside of triangle ABC.

Points E and F disappear because there altitudes don’t intersect the sides of ABC anymore. So segments CF and BE don’t exist now. With no longer having segments CF or BE, the two proven equations do not apply to the obtuse version of ABC.