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.

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.

- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- 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)

- 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

- 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 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)

- 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

- 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

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