Printable instructions for finding the orthocenter of a triangle with compass and straightedge or ruler

Constructing the orthocenter of a triangle

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See also the animated version.

After doing this	Your work should look like this
We start with the triangle ABC.
1. Set the compasses' width to the length of a side of the triangle. Any side will do, but the shortest works best.
2. With the compasses on B, one end of that line, draw an arc across the opposite side. Label this point F
3. Repeat for the other end of the line, C. Label this point P.
Note If you find you cannot draw these arcs on the opposite sides, the orthocenter is outside the triangle. See note below
What we do now is draw two altitudes. This is the same process as constructing a perpendicular to a line through a point.
4. With the compasses on B, set the compasses' width to more than half the distance to P.
5. From B and P, draw two arcs that intersect, creating point Q.
6. Use a straightedge to draw a line from C to Q. The part of this line inside the triangle forms an altitude of the triangle.
Now we repeat the process to create a second altitude.
7. With the compasses on C, set the compasses' width tomore than half the distance to F.
8. From C and F, draw two arcs that intersect, creating point E.
9. Use a straightedge to draw a line from B to E. The part of this line inside the triangle forms an altitude of the triangle.
10. Done. The point where the two altitudes intersect is the orthocenter of the triangle. (You may need to extend the altitude lines so they intersect if the orthocenter is outside the triangle)
Optional Step 11. Repeat steps 7,8,9 on the third side of the triangle. This will help convince you that all three altitudes do in fact intersect at a single point. But two altitudes are enough to find that point.

After doing this

Your work should look like this

We start with the triangle ABC.

Geometry construction with compass and straightedge or ruler or ruler

1. Set the compasses' width to the length of a side of the triangle. Any side will do, but the shortest works best.

2. With the compasses on B, one end of that line, draw an arc across the opposite side. Label this point F

3. Repeat for the other end of the line, C. Label this point P.

*Note If you find you cannot draw these arcs on the opposite sides, the orthocenter is outside the triangle. See note below*

What we do now is draw two altitudes. This is the same process as constructing a perpendicular to a line through a point.

4. With the compasses on B, set the compasses' width to more than half the distance to P.

5. From B and P, draw two arcs that intersect, creating point Q.

6. Use a straightedge to draw a line from C to Q. The part of this line inside the triangle forms an altitude of the triangle.

Now we repeat the process to create a second altitude.

7. With the compasses on C, set the compasses' width tomore than half the distance to F.

8. From C and F, draw two arcs that intersect, creating point E.

9. Use a straightedge to draw a line from B to E. The part of this line inside the triangle forms an altitude of the triangle.

10. Done. The point where the two altitudes intersect is the orthocenter of the triangle. (You may need to extend the altitude lines so they intersect if the orthocenter is outside the triangle)

Optional Step 11.
Repeat steps 7,8,9 on the third side of the triangle. This will help convince you that all three altitudes do in fact intersect at a single point. But two altitudes are enough to find that point.

Constructing the orthocenter of a triangle

Other constructions pages on this site

Lines

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions