In or Out?
Each triangle has three altitudes. The diagram above illustrates three red altitudes of a particular triangle. Given any triangle, do the altitudes lie on a side, on the inside, or on the outside of the triangle? How do your results depend on different types of triangles?

Extensions 
Construct the orthocenter, the location where the lines containing the altitudes intersect. How does the location of the orthocenter depend on the type of triangle?
Why do you think this point of intersection is called the "orthocenter"?
Submit your idea for an investigation to InterMath.
