
Orthocenter of a Triangle
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.
Printable stepbystep instructions
The above animation is available as a
printable stepbystep instruction sheet, which can be used for making handouts
or when a computer is not available.
Proof
 Q.E.D
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.
Try it yourself
Click here for a printable worksheet containing two triangle orthocenter problems.
When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.
Other constructions pages on this site
Lines
Angles
Triangles
Right triangles
Triangle Centers
Circles, Arcs and Ellipses
Polygons
NonEuclidean constructions
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