The Orthocenter Calculator finds the point where a triangle’s three altitudes intersect. Enter the three vertex coordinates, and it returns the orthocenter’s (x, y) location using a coordinate-geometry formula.
This page also explains what the orthocenter means, how it relates to altitudes, and how to interpret results for acute, right, and obtuse triangles.
What Is the Orthocenter?
The orthocenter of a triangle is the common intersection point of the triangle’s three altitudes. An altitude is a line through a vertex that is perpendicular to the opposite side.
Depending on the triangle’s shape:
- Acute triangle: the orthocenter lies inside the triangle.
- Right triangle: the orthocenter is at the right-angle vertex.
- Obtuse triangle: the orthocenter lies outside the triangle.
Coordinate Formula (Core Concept)
When you know the vertices in the coordinate plane, you can compute the orthocenter directly. Let the triangle’s vertices be:
- A(x1, y1)
- B(x2, y2)
- C(x3, y3)
To compute the orthocenter H(xh, yh), use the intersection of two altitude lines. The calculator uses a robust method based on determinants.
Determinant Helpers
Define the following 2×2 determinants:
| Expression | Meaning |
|---|---|
| D | (x1(y2−y3) + x2(y3−y1) + x3(y1−y2)) |
| Dx | ( (x1^2+y1^2)(y2−y3) + (x2^2+y2^2)(y3−y1) + (x3^2+y3^2)(y1−y2) ) |
| Dy | ( (x1^2+y1^2)(x3−x2) + (x2^2+y2^2)(x1−x3) + (x3^2+y3^2)(x2−x1) ) |
Then the orthocenter coordinates are:
- xh = Dx / D
- yh = Dy / D
Important: If D = 0, the points are collinear (not a valid triangle), so the orthocenter is undefined.
How the Inputs Map to Altitudes
Each altitude is perpendicular to a side. In coordinate geometry, perpendicular lines have slopes that are negative reciprocals (when slopes are defined). The determinant method avoids special cases like vertical or horizontal sides by working algebraically.
In practical terms, the calculator is effectively finding where two altitudes meet, which must also be where the third altitude meets (for a non-degenerate triangle).
Using the Orthocenter Calculator (Step-by-Step)
- Enter the coordinates of A(x1, y1), B(x2, y2), and C(x3, y3).
- Click Calculate.
- Read the output values for H(x, y).
- If your triangle is degenerate (points on a line), the calculator will show an error.
Units don’t affect the math: if your coordinates are in centimeters, the orthocenter will also be in centimeters. The calculator treats x and y units consistently.
Practical Examples
Example 1: Acute Triangle (Orthocenter Inside)
Suppose:
- A(1, 1)
- B(5, 2)
- C(3, 6)
For an acute triangle like this, the orthocenter falls inside the triangle. The calculator returns the exact intersection point of the altitudes, giving you the coordinates directly.
Example 2: Right Triangle (Orthocenter at a Vertex)
Suppose the triangle is right-angled at A:
- A(0, 0)
- B(4, 0)
- C(0, 3)
The orthocenter is exactly at the right-angle vertex, so H(0, 0). The calculator will output that point because the altitudes intersect there.
Common Mistakes to Avoid
- Mixing up x and y: coordinates must be entered as (x, y) for each vertex.
- Using collinear points: three points on a line do not form a triangle; the orthocenter is undefined.
- Assuming the orthocenter is always inside: it can be outside for obtuse triangles.
Frequently Asked Questions
What is an orthocenter in simple terms?
The orthocenter is the point where all three triangle altitudes meet. Each altitude is a line through a vertex perpendicular to the opposite side. In an acute triangle it lies inside, in a right triangle it matches the right-angle vertex, and in an obtuse triangle it lies outside.
How do I find the orthocenter using coordinates?
With vertex coordinates A(x1,y1), B(x2,y2), and C(x3,y3), compute the orthocenter intersection of two altitudes. A fast approach uses determinant formulas to calculate H(xh,yh). If the determinant denominator is zero, the points are collinear and the triangle is invalid.
Can the orthocenter be outside the triangle?
Yes. For obtuse triangles, the altitudes intersect outside the triangle’s interior. The orthocenter still exists and is the same intersection point of the altitude lines; it just falls in a region outside the triangle. Your calculator will output coordinates that reflect that location.
What happens for a right triangle?
In a right triangle, two sides are perpendicular, so the altitude from the right-angle vertex is the side itself. That means the orthocenter equals the right-angle vertex. For coordinates, the orthocenter coordinates match the vertex with the 90° angle.
Why does my calculator show an error?
An error usually means the three points do not form a valid triangle. If the vertices are collinear, the altitudes do not intersect at a single orthocenter point. Check that you entered correct coordinates for A, B, and C and that they are not on the same straight line.
Bottom Line
The Orthocenter Calculator gives the orthocenter coordinates of a triangle from vertex coordinates in one step. Enter (x, y) values for A, B, and C, and the calculator returns H(x, y) using altitude intersection geometry.
