Orthocenter

Orthocenter

The point at which the three (possibly extended) altitudes of a triangle intersect. The orthocenter is one of the centers of a triangle.

Orthocenter of an Acute Triangle
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Orthocenter of an Obtuse Triangle

Worked Example

Problem: Find the orthocenter of the triangle with vertices A(0, 0), B(6, 0), and C(2, 4).

Step 1: Find the slope of side BC. The altitude from A is perpendicular to BC.

\text{slope of } BC = \frac{4 - 0}{2 - 6} = \frac{4}{-4} = -1

Step 2: The altitude from A is perpendicular to BC, so its slope is the negative reciprocal of −1, which is 1. Since it passes through A(0, 0):

y = x

Step 3: Find the slope of side AC. The altitude from B is perpendicular to AC.

\text{slope of } AC = \frac{4 - 0}{2 - 0} = 2

Step 4: The altitude from B has slope −1/2 (the negative reciprocal of 2) and passes through B(6, 0):

y - 0 = -\tfrac{1}{2}(x - 6) \quad\Longrightarrow\quad y = -\tfrac{1}{2}x + 3

Step 5: Set the two altitude equations equal to find their intersection:

x = -\tfrac{1}{2}x + 3 \quad\Longrightarrow\quad \tfrac{3}{2}x = 3 \quad\Longrightarrow\quad x = 2,\; y = 2

Answer: The orthocenter is at (2, 2).

Frequently Asked Questions

Where is the orthocenter located for different types of triangles?

For an acute triangle, the orthocenter lies inside the triangle. For a right triangle, it sits exactly at the vertex of the right angle. For an obtuse triangle, it falls outside the triangle — you must extend the altitudes beyond the triangle's sides to find where they meet.

Is the orthocenter always inside the triangle?

No. It is only inside when the triangle is acute. In an obtuse triangle the orthocenter is outside, and in a right triangle it coincides with the vertex at the right angle.

Orthocenter vs. Centroid

The orthocenter is the intersection of the three altitudes, while the centroid is the intersection of the three medians. The centroid always lies inside the triangle and divides each median in a 2 : 1 ratio. The orthocenter, by contrast, can lie outside the triangle when it is obtuse. Both are classical triangle centers, but they coincide only in an equilateral triangle.

Why It Matters

The orthocenter is one of the four classical triangle centers (along with the centroid, circumcenter, and incenter) and plays a key role in advanced geometry. All four centers coincide in an equilateral triangle, but in any other triangle they are distinct. The orthocenter also lies on the Euler line, which connects it to the centroid and circumcenter — a relationship that appears frequently in geometry proofs and competition problems.

Common Mistakes

Mistake: Confusing an altitude with a median or a perpendicular bisector.

Correction: An altitude drops from a vertex perpendicular to the opposite side. A median connects a vertex to the midpoint of the opposite side. A perpendicular bisector is perpendicular to a side at its midpoint but does not necessarily pass through a vertex. Make sure you draw the correct line segment when constructing the orthocenter.

Mistake: Assuming the orthocenter is always inside the triangle.

Correction: The orthocenter lies inside only for acute triangles. For obtuse triangles it is outside, and for right triangles it is at the right-angle vertex. When working with an obtuse triangle, extend the altitudes beyond the sides to locate the orthocenter.

Related Terms

Altitude of a Triangle — The three lines whose intersection defines the orthocenter
Triangle — The shape in which the orthocenter is defined
Centers of a Triangle — Family of special points including the orthocenter
Centroid — Intersection of medians; another triangle center
Circumcenter — Intersection of perpendicular bisectors; lies on Euler line
Incenter — Intersection of angle bisectors; another triangle center
Perpendicular — Key geometric relationship used to construct altitudes