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
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
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more
Triangles
- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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