This page shows how to construct the orthocenter of a triangle with compass and straightedge or ruler. For a more, see orthocenter of a triangle. The orthocenter is the point where all three altitudes of the triangle intersect. An altitude is a line which passes through a vertex of the triangle and is perpendicular to the opposite side. There are therefore three altitudes in a triangle. It works using the construction for a perpendicular through a point to draw two of the altitudes, thus location the orthocenter.
*Note If you find you cannot draw the arcs in steps 2 and 3, the orthocenter lies outside the triangle. See
Orthocenter of a triangle.
To solve the problem, extend the opposite side until you can draw the arc across it. (See diagram right). Then proceed as usual.
The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.
|1||CQ is perpendicular to AB||By construction. See Perpendicular to a line from an external point with compass and straightedge for method and proof.|
|2||CQ is an altitude of the triangle ABC||An altitude of a triangle is a line segment through a vertex and perpendicular to the opposite side.|
|3||BE is perpendicular to AC||By construction. See Perpendicular to a line from an external point with compass and straightedge for method and proof.|
|4||BE is an altitude of the triangle ABC||An altitude of a triangle is a line segment through a vertex and perpendicular to the opposite side.|
|5||O is the orthocenter of the triangle ABC||The orthocenter of a triangle is the point where its altitudes intersect|
The three altitudes all intersect at the same point so we only need two to locate it. The proof for the third one is similar to the above.