What do you notice about these altitudes you drew?[br][br]If you move A, B, or C around, what changes? What stays the same?[br][br]Move A,B, or C so that the triangle is acute and describe what you see. Make the triangle right, what do you see? Make the triangle obtuse, what do you see?
We see that the three altitudes intersect at the same place! This is called a [b]point of concurrency[/b] when three or more lines intersect at a single point; it doesn't happen very often so this is pretty special. Take three pencils and drop them onto a table, did all three cross at the same place? Odds are no, the chances of this happening are very low but they happened here.[br][br]This point is called the o[b]rthocenter, [/b]and it exists in every triangle. When you moved the blue points did you notice that the three lines still crossed? The point moved but they still intersected and this will always happen.[br][br]When the triangle is acute, the orthocenter is inside the triangle. When the triangle is right, the orthocenter touches the triangle, and when the triangle is obtuse the orthocenter is outside the triangle![br][br]The interesting trait of the orthocenter is that if you pick any two points on the triangle with the orthocenter and make a triangle with those points, the remaining point is the new orthocenter. Play around with your figure and prove to yourself this is true.