Orthocenter - Definition, Properties, and Examples
Orthocenter - Definition, Properties, and Examples
TL;DR
The orthocenter is the single point where a triangle's three altitudes meet. It sits inside an acute triangle, on the right-angle vertex of a right triangle, and outside an obtuse triangle. This article shows how to find it by hand and on the coordinate plane, and how it differs from the circumcenter, incenter, and centroid.
What Is The Orthocenter of A Triangle?
The orthocenter of a triangle is the point where its three altitudes intersect, usually written as HHH. An altitude is a line segment drawn from a vertex straight down to the opposite side, meeting that side at a right angle (90°). Every triangle has exactly three altitudes, one from each vertex - and all three pass through the same single point.
That "same single point" is not obvious. There is no reason three separately-drawn perpendiculars should agree on where to cross. But they always do, for every triangle. That guaranteed meeting is called concurrency, and the orthocenter is one of a triangle's four classic centres.
How Do You Find The Orthocenter?
You never actually need all three altitudes. Because the three altitudes are guaranteed to be concurrent, finding where any two of them cross gives the orthocenter - the third altitude will pass through the same point automatically.
By hand, the steps are:
- Draw one altitude - from a vertex, drop a perpendicular to the opposite side.
- Draw a second altitude - from a different vertex, drop a perpendicular to its opposite side.
- Mark where they cross. That intersection is the orthocenter HHH.
On the coordinate plane, you turn "perpendicular" into slopes. The altitude from a vertex is perpendicular to the opposite side, so its slope is the negative reciprocal of that side's slope. If a side has slope mmm, the altitude to it has slope −1/m. Find the equations of two altitudes and solve them together - the solution is HHH.
Where does the orthocenter sit?
Its position depends entirely on the triangle's shape:
- Acute triangle (all angles < 90°) - orthocenter lies inside.
- Right triangle (one 90° angle) - orthocenter sits exactly on the right-angle vertex, because the two legs are themselves altitudes.
- Obtuse triangle (one angle > 90°) - orthocenter lies outside the triangle, and two of the altitudes must be extended beyond the sides to reach it.
How Is the Orthocenter Different From the Circumcenter, Incenter, and Centroid?
Every triangle has four famous centres, and each comes from a different set of concurrent lines. Mixing them up is the most common source of error, so hold the distinctions clearly.
| Centre | Built from | Special property |
|---|---|---|
| Orthocenter (HHH) | The three altitudes | Inside/on/outside by triangle type |
| Circumcenter (OOO) | The three perpendicular bisectors of the sides | Centre of the circle through all three vertices |
| Incenter (III) | The three angle bisectors | Centre of the circle inside, touching all three sides |
| Centroid (GGG) | The three medians (vertex to midpoint of opposite side) | Balance point; divides each median 2:1 |
Two facts make the set easier to hold together. In an equilateral triangle, all four centres land on the exact same point. And in any triangle, the orthocenter HHH, centroid GGG, and circumcenter OOO always lie on one straight line called the Euler line, with the centroid sitting between the other two in a fixed HG:GO=2:1 ratio.
Examples of the Orthocenter
Each example is worked step by step. The problem statement is in bold; the working is not.
Example 1
In which region of an acute triangle does the orthocenter lie?
An acute triangle has all three angles less than 90°.
By the position rule, the orthocenter of an acute triangle lies inside the triangle.
Final answer: inside the triangle.
Example 2
A student is asked for the orthocenter of a right triangle with the right angle at C. They start solving two altitude equations from scratch. Is there a faster way, and is the setup even right?
The instinct is to treat every triangle the same: write slopes, find negative reciprocals, solve simultaneously. That works, but for a right triangle it hides something.
Watch what the general method gives. Take the right triangle with vertices A(0,0), B(4,0), C(0,3), right angle at C... except the right angle is actually at A here, since AB lies along the x-axis and AC along the y-axis, meeting at A(0,0) at 90°.
That mislabelling is the trap. Before grinding through algebra, look at the shape. The two legs of a right triangle are altitudes: leg AB is perpendicular to leg AC, so each leg is the altitude from its far vertex. Both legs meet at the right-angle vertex.
So the orthocenter sits exactly on the right-angle vertex - here, A(0,0). No simultaneous equations needed.
Final answer: The orthocenter of a right triangle is the right-angle vertex itself. Read the triangle type first; the algebra confirms what the shape already tells you.
Example 3
Find the orthocenter of the triangle with vertices A(0, 0), B(6, 0), and C(2, 4).
Find two altitudes.
Altitude from C to side AB: side AB lies along the x-axis (slope 0), so its altitude is vertical through C(2,4): x=2
Altitude from A to side BC: side BC goes from B(6,0) to C(2,4), slope = (4−0)/(2−6) = 4/-4 = -1. The altitude is perpendicular, slope = +1, through A(0,0): y=x
Solve the two together: substitute x=2 into y=x: y=2
Final answer: The orthocenter is at (2,2).
Example 4
A triangle has one angle of 120°. Where is its orthocenter?
A 120° angle is greater than 90°, so this is an obtuse triangle.
The orthocenter of an obtuse triangle lies outside the triangle; two of its altitudes must be extended beyond the sides to reach the meeting point.
Final answer: outside the triangle.
Example 5
Find the orthocenter of the triangle with vertices A(1, 1), B(5, 1), C(3, 5).
Altitude from C to AB: AB is horizontal (slope 0), altitude is vertical through C(3,5): x=3
Altitude from A to BC: BC runs from B(5,1) to C(3,5), slope = (5−1)/(3−5) = 4/-2 = -2. Perpendicular slope = +1/2, through A(1,1): y−1=(1/2)(x−1)
Substitute x=3: y−1=(1/2)(2)=1
Final answer: The orthocenter is at (3,2).
Example 6
Verify that the third altitude of the triangle in Example 5 also passes through (3, 2).
The third altitude is from B(5,1) to side AC. Side AC runs from A(1,1) to C(3,5), slope = (5−1)/(3−1) = 2. Perpendicular slope = -1/2, through B(5,1): y−1=(-1/2)(x−5)
Test the point (3,2): 2−1=(-1/2)(-2), which confirms.
Final answer: The third altitude passes through (3,2), confirming the concurrency of all three altitudes.
Why the Orthocenter Matters — "Three lines forced to agree"
The orthocenter earns its keep as a first, clean lesson in concurrency - the idea that certain constructions in a triangle are guaranteed to meet at one point, not merely likely to. That guarantee is a theorem, not luck, and it is the reason triangle geometry is so tightly interlocked.
Where that idea does real work:
- The Euler line - the discovery that the orthocenter, centroid, and circumcenter always line up is one of the surprises of classical geometry. Three centres built from three completely different rules, yet always collinear.
- Coordinate geometry and computer graphics - turning "the altitudes meet here" into an (x,y) solution is exactly the kind of intersection-of-perpendiculars computation that rendering engines and CAD systems run constantly.
- The nine-point circle - the orthocenter is one of the anchor points for a circle that passes through nine special points of any triangle.
What Are the Most Common Mistakes With the Orthocenter?
Mistake 1: Confusing the orthocenter with the centroid
Where it slips in: Whenever a question names a triangle centre without spelling out which lines build it.
Don't do this: Using medians when the question asks for the orthocenter.
The correct way: The orthocenter comes from altitudes; the centroid comes from medians. Read which one the construction actually uses before computing.
Mistake 2: Expecting the orthocenter to be inside every triangle
Where it slips in: In obtuse-triangle problems, where a student solves correctly but then rejects a valid answer for landing outside the triangle.
Don't do this: Assuming a coordinate answer is wrong because HHH falls outside the shape.
Mistake 3: Using the side's slope instead of the perpendicular slope
Where it slips in: Setting up altitude equations.
Don't do this: Giving the altitude the same slope as the side it drops to.
Conclusion
- The orthocenter is the point where a triangle's three altitudes meet.
- Find it by solving any two altitude equations; the third passes through the same point.
- Its position depends on the triangle: inside (acute), on the right-angle vertex (right), outside (obtuse).
- It differs from the circumcenter, incenter, and centroid.