Orthocenter Calculator

Understanding the Orthocenter of a Triangle

The orthocenter is one of the four classical centers of a triangle, formed at the point where all three altitudes meet. An altitude is a line drawn from a vertex straight down to the opposite side at a right angle. While the concept sounds straightforward, the orthocenter has rich geometric properties that connect it to many other areas of mathematics, from Euler lines to reflection theorems and coordinate geometry.

Unlike the centroid, which always sits inside the triangle, the orthocenter can move to different positions depending on the triangle's shape. For acute triangles it stays inside, for right triangles it lands on a vertex, and for obtuse triangles it moves outside entirely. This behavior makes the orthocenter especially interesting for students and professionals working with triangle geometry.

How to Find the Orthocenter Step by Step

Finding the orthocenter from vertex coordinates requires a systematic approach. Given three vertices A, B, and C, the process involves calculating slopes, forming perpendicular lines, and solving a system of equations. Here is how it works in practice.

First, pick two sides of the triangle and calculate their slopes. The slope of a line through two points (x1, y1) and (x2, y2) is given by m = (y2 - y1) / (x2 - x1). Next, find the perpendicular slope for each side by taking the negative reciprocal: if the side has slope m, the altitude perpendicular to it has slope -1/m. Then write the equation of each altitude using point-slope form, passing through the opposite vertex. Finally, solve the two altitude equations simultaneously to find the intersection point. That point is the orthocenter.

For example, with vertices at A(0, 0), B(6, 0), and C(2, 5), the slope of BC is (0 - 5) / (6 - 2) = -1.25. The altitude from A has slope 1/1.25 = 0.8 and passes through A, giving y = 0.8x. Following the same process for a second altitude and solving yields the orthocenter coordinates. Our calculator handles all of this automatically, including special cases like vertical and horizontal sides.

What Is an Altitude of a Triangle?

An altitude of a triangle is a perpendicular line segment from a vertex to the line containing the opposite side. Every triangle has three altitudes, one from each vertex. The point where an altitude meets the opposite side (or its extension) is called the foot of the altitude. In an acute triangle, all three feet lie on the sides of the triangle. In an obtuse triangle, two of the feet land on extensions of the sides, outside the triangle boundary.

Altitudes serve several purposes in geometry. They are used to calculate the area of a triangle (area = 0.5 * base * height, where the height is the length of the altitude to that base). They also play a role in defining the orthocentric system, where the orthocenter of one triangle becomes a vertex of related triangles. The three altitudes always meet at a single point, which is a consequence of a fundamental theorem in Euclidean geometry.

Orthocenter Position by Triangle Type

The location of the orthocenter is directly linked to the largest angle in the triangle. In an acute triangle, where all angles measure less than 90 degrees, every altitude falls inside the triangle, so their intersection also lies inside. This is the most intuitive case, as you can visualize each perpendicular line dropping from a vertex to the interior of the opposite side.

In a right triangle, the two legs are each perpendicular to the other. That means the two legs are themselves altitudes, and they intersect at the vertex of the right angle. The third altitude also passes through this vertex, confirming that the orthocenter coincides with the right-angle vertex. This is easy to verify by entering a right triangle such as A(0, 0), B(4, 0), C(0, 3) into our calculator.

For obtuse triangles, the orthocenter lies outside the triangle on the side of the obtuse angle. The altitudes from the two acute vertices must extend beyond the opposite side to reach the perpendicular foot, and these extended lines meet at a point outside the triangle. Try the obtuse example in our calculator to see how the dashed altitude lines clearly extend beyond the triangle edges.

The Four Triangle Centers Compared

A triangle has four classical centers, each defined by a different set of special lines. The orthocenter (H) is the intersection of the three altitudes. The centroid (G) is where the three medians meet and represents the triangle's center of mass. The circumcenter (O) is the intersection of the three perpendicular bisectors and serves as the center of the circle that passes through all three vertices. The incenter (I) is where the three angle bisectors meet and is the center of the largest circle that fits inside the triangle.

Each center has unique properties. The centroid always lies inside the triangle and divides each median in a 2:1 ratio. The circumcenter can be inside, on, or outside the triangle depending on whether it is acute, right, or obtuse. The incenter is always inside the triangle. The orthocenter's position varies as described above. Our calculator computes all four centers simultaneously so you can compare their coordinates side by side.

The Euler Line

One of the most elegant results in triangle geometry is the Euler line, discovered by Leonhard Euler in 1765. This theorem states that the orthocenter, centroid, and circumcenter of any non-equilateral triangle are collinear, meaning they all lie on a single straight line. Furthermore, the centroid divides the segment from the orthocenter to the circumcenter in a fixed 2:1 ratio. That is, the distance from the orthocenter to the centroid is always twice the distance from the centroid to the circumcenter.

In an equilateral triangle, all four classical centers coincide at the same point, so the Euler line is undefined. For all other triangles, the Euler line provides a powerful relationship between these centers. You can verify the Euler line property by computing the three centers with our calculator and checking that they are aligned.

The Orthocentric System

A fascinating property of the orthocenter is that it creates an orthocentric system with the triangle's vertices. If H is the orthocenter of triangle ABC, then A is the orthocenter of triangle HBC, B is the orthocenter of triangle AHC, and C is the orthocenter of triangle ABH. In other words, any of the four points is the orthocenter of the triangle formed by the other three. This symmetric relationship reveals deep structure in the geometry of altitudes.

This property has practical consequences. It means that the orthocenter is not just a derived point but a fundamental part of the four-point configuration. The orthocentric system also relates to the nine-point circle, another classical construction that passes through the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from the vertices to the orthocenter.

Applications in Mathematics and Beyond

The orthocenter appears in numerous mathematical proofs and problems. In competition mathematics, orthocenter properties are frequently tested because they connect many geometric concepts: perpendicularity, concurrence, collinearity, and circle theorems. Problems involving the nine-point circle, the Euler line, or reflections of the orthocenter over the sides often appear in mathematical olympiads.

In coordinate geometry courses, finding the orthocenter is a classic exercise that reinforces skills in slope calculation, perpendicular lines, and solving systems of equations. The problem combines multiple concepts into one task, making it an excellent assessment of a student's understanding of analytic geometry. Our step-by-step solution shows exactly how each calculation flows into the next, which can help students check their own work or learn the method from scratch.

Beyond pure mathematics, the concept of perpendicular intersections appears in engineering, computer graphics, and geographic information systems. Triangulation algorithms used in surveying and mesh generation rely on understanding how perpendicular constructions behave within triangles. While these applications may not reference the orthocenter by name, the underlying geometry is the same.