Orthocenter Calculator
Calculate the orthocenter of a triangle from vertex coordinates. Find altitude intersection, triangle type, angles, and side lengths instantly.
Formula
Orthocenter = Intersection of altitudes (perpendicular from vertex to opposite side)
The orthocenter is found by computing two altitudes (lines from a vertex perpendicular to the opposite side) and finding their intersection. Each altitude slope is the negative reciprocal of the slope of the opposite side.
Worked Examples
Example 1: Orthocenter of an Acute Triangle
Problem:Find the orthocenter of the triangle with vertices A(0, 6), B(-3, 0), C(3, 0).
Solution:Slope of BC = (0 - 0)/(3 - (-3)) = 0 (horizontal line) Altitude from A perpendicular to BC is vertical: x = 0 Slope of AC = (0 - 6)/(3 - 0) = -2 Altitude from B perpendicular to AC has slope = 1/2 Equation: y - 0 = 0.5(x - (-3)) => y = 0.5x + 1.5 At x = 0: y = 1.5 Orthocenter = (0, 1.5)
Result:Orthocenter: (0, 1.5) — Inside the triangle (acute triangle)
Example 2: Orthocenter of an Obtuse Triangle
Problem:Find the orthocenter of the triangle with vertices A(0, 0), B(5, 0), C(1, 1).
Solution:Slope of BC = (1 - 0)/(1 - 5) = -0.25 Altitude from A: slope = 4, equation: y = 4x Slope of AC = (1 - 0)/(1 - 0) = 1 Altitude from B: slope = -1, equation: y = -(x - 5) = -x + 5 Intersection: 4x = -x + 5 => 5x = 5 => x = 1, y = 4 Orthocenter = (1, 4)
Result:Orthocenter: (1, 4) — Outside the triangle (obtuse triangle)
Frequently Asked Questions
What is the orthocenter of a triangle?
The orthocenter of a triangle is the single point where all three altitudes of the triangle intersect. An altitude is a line segment drawn from any vertex of the triangle perpendicular to the opposite side, or to the extension of that opposite side. Every non-degenerate triangle has exactly one orthocenter. The orthocenter is one of the four classical triangle centers alongside the centroid, circumcenter, and incenter. Unlike the centroid which always lies inside the triangle, the orthocenter can be inside, outside, or on the triangle depending on whether the triangle is acute, obtuse, or right respectively.
How do you calculate the orthocenter of a triangle?
To find the orthocenter, you need to determine the intersection point of at least two altitudes of the triangle. First, calculate the slope of one side of the triangle. The altitude from the opposite vertex has a slope that is the negative reciprocal of the side slope. Write the equation of this altitude line using the point-slope form with the vertex coordinates. Repeat the process for a second altitude using a different side and its opposite vertex. Then solve the two altitude line equations simultaneously to find their intersection point. That intersection is the orthocenter. The third altitude will automatically pass through the same point.
Where is the orthocenter located for different triangle types?
The location of the orthocenter depends entirely on the type of triangle. For an acute triangle, where all three interior angles are less than 90 degrees, the orthocenter lies strictly inside the triangle. For a right triangle, the orthocenter is located exactly at the vertex where the right angle is formed, because the two legs of the right triangle are themselves altitudes. For an obtuse triangle, where one angle exceeds 90 degrees, the orthocenter lies outside the triangle on the side opposite to the obtuse angle. This property makes the orthocenter unique among triangle centers.
What is the Euler line and how does the orthocenter relate to it?
The Euler line is a remarkable straight line that passes through several important triangle centers including the orthocenter, the centroid, and the circumcenter. The centroid always divides the segment from the orthocenter to the circumcenter in a ratio of two to one, with the centroid being closer to the circumcenter. This relationship was discovered by the Swiss mathematician Leonhard Euler in the 18th century. The Euler line demonstrates a deep geometric relationship between these triangle centers. In equilateral triangles, all three points coincide, so technically no unique Euler line exists for equilateral triangles.
What is the difference between orthocenter, centroid, circumcenter, and incenter?
These four points are the classical triangle centers but they have different definitions and properties. The centroid is the intersection of the three medians, which connect each vertex to the midpoint of the opposite side, and it always lies inside the triangle. The circumcenter is the intersection of the perpendicular bisectors of the three sides and is equidistant from all three vertices. The incenter is where the three angle bisectors meet and is equidistant from all three sides. The orthocenter is where the three altitudes intersect. Only the centroid and incenter are guaranteed to lie inside the triangle for all triangle shapes.
What is the orthic triangle, and how is it connected to the orthocenter?
The orthic triangle is formed by the three feet of the altitudes, the points where each altitude meets the opposite side. For an acute triangle the orthocenter of the original triangle is exactly the incenter of the orthic triangle, so the three altitudes bisect the interior angles of the orthic triangle while the sides of the original triangle act as its external bisectors. The orthic triangle also solves Fagnano problem: of all triangles inscribed in a given acute triangle, it has the smallest perimeter. Both results fail for obtuse triangles, where the altitude feet fall on side extensions.
Why is each vertex the orthocenter of the triangle formed by the other two and H?
A triangle and its orthocenter form what is called an orthocentric system, a set of four points in which any one is the orthocenter of the triangle made by the remaining three. If H is the orthocenter of triangle ABC, then A is the orthocenter of triangle HBC, and likewise for B and C. The reason is that perpendicularity is symmetric: the line AH is perpendicular to BC no matter which of the two triangles you regard it as an altitude of. All four triangles in such a system also share the same nine-point circle.
Is there a vector formula that gives the orthocenter without solving two line equations?
Yes. Take the circumcenter O as the origin, so that the position vectors of the three vertices all have length equal to the circumradius. Then the orthocenter is simply H = A + B + C, the vector sum of the three vertices. This identity immediately implies the Euler line relation, since the centroid is (A + B + C)/3 and therefore lies one third of the way from O to H. In barycentric coordinates relative to the triangle, the orthocenter is tan A : tan B : tan C, which is undefined for a right triangle because one tangent blows up, matching the fact that H then sits on a vertex.
Why do three collinear points have no orthocenter?
Collinear points form a degenerate triangle with zero area, and the altitudes from each point are then three parallel perpendicular lines that never meet in a single point. Orthocenter Calculator tests for that case using the signed area expression from the shoelace formula, x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2), and returns no result when it is effectively zero. In practice near-collinear vertices are the more dangerous case, because the altitudes intersect at a huge distance and small rounding errors in the coordinates move the computed orthocenter a long way.
What is the nine-point circle, and how does it involve the orthocenter?
Every triangle has a circle passing through nine notable points: the three side midpoints, the three feet of the altitudes, and the three midpoints of the segments joining the orthocenter to each vertex. Its centre is the midpoint of the segment from the orthocenter to the circumcenter, so it lies on the Euler line, and its radius is exactly half the circumradius. Because three of the nine points are altitude feet and three more are defined from H, the orthocenter is central to the construction, and it is one of the classic results connecting the triangle centres.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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