**Write-Up #8**

**Altitudes and Orthocenters**

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

**Prove: **

**First, construct the orthocenter H of triangle
ABC.**

**Next, let points D, E, and F be the feet
of the perpendiculars from A, B, and C respectfully.**

**From the picture below, we can see that
triangle ABC is composed of 3 smaller triangles, **.

**Since the 3 smaller triangles compose triangle
ABC, we can represent the area of triangle ABC in terms of the
areas of the 3 smaller triangles.**

**In other words, **.

**We can divide both sides by .**

**The area of a triangle is 1/2 * base * height.
Substitute this formula into each of the triangles in our above
equation.**

**Canceling factors in the numerators and
denominators of each term, we find:**

**Rearranging terms:**

**Q.E.D.**