How To Construct (draw) The Orthocenter Of A Triangle

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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 step-by-step instructions

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.

Proof

Argument	Reason
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

- 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

• List of printable constructions worksheets

Lines

• Introduction to constructions
• Copy a line segment
• Sum of n line segments
• Difference of two line segments
• Perpendicular bisector of a line segment
• Perpendicular at a point on a line
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Right triangles

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• Right Triangle, given one leg and one angle (LA)

Triangle Centers

• Triangle incenter
• Triangle circumcenter
• Triangle orthocenter
• Triangle centroid

Circles, Arcs and Ellipses

• Finding the center of a circle
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Polygons

• Square given one side
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Non-Euclidean constructions

• Construct an ellipse with string and pins
• Find the center of a circle with any right-angled object

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