Question

Why is it that the orthocenter of an obtuse triangle must lie on the outside of the triangle?

Answer 1

The orthocenter of a triangle is the intersection of all the altitudes in said triangle. These altitudes can be formed by drawing a line from each corner to the opposite side, and making sure that line forms a right angle with the side.

When trying to draw the altitudes in an obtuse triangle, you'll find that you're only able to successfully get a right angle with one side (opposite of the obtuse angle). The other two sides you'll have to extend out so a right angle can be formed when the lines are drawn.

When you do this, you'll have three lines that are the altitudes of the triangle, but these lines won't meet, so you'll have to extend the altitudes out a little bit so they all intersect.