Orthocenter Calculator
Calculate the orthocenter of any triangle from its three vertex coordinates. Free online geometry tool finds orthocenter, classifies triangle type, and computes side lengths and area.
About This Calculator
What Is the Orthocenter of a Triangle?
The orthocenter of a triangle is the unique point where all three altitudes of the triangle intersect. An altitude is a line segment drawn from a vertex perpendicular to the opposite side (or its extension). In every triangle, the three altitudes are concurrent, meaning they always meet at a single point -- the orthocenter, commonly denoted as H.
This free online orthocenter calculator finds the orthocenter coordinates from three vertex coordinates using coordinate geometry. Enter the x and y coordinates for vertices A, B, and C, and the calculator will compute the orthocenter (hx, hy), classify the triangle as acute, right, or obtuse, display all three side lengths (AB, BC, AC), and calculate the triangle's area using the shoelace formula. An interactive chart visualizes the triangle with its orthocenter and a bar chart shows the side length comparison.
How the Orthocenter Is Calculated
The calculator uses the altitude-line method: first it computes the slope of each side, then determines the perpendicular slope to get the equation of the altitude from the opposite vertex. Two altitude equations are solved as a system of linear equations to find their intersection point -- the orthocenter. Special cases like vertical sides (infinite slope) and horizontal sides (zero slope) are handled separately using horizontal and vertical line equations.
Understanding Triangle Classification
The calculator classifies triangles based on squared side lengths: if the square of the longest side equals the sum of squares of the other two, the triangle is right (Pythagorean theorem). If it exceeds the sum, the triangle is obtuse (one angle exceeds 90 deg). If it is less, all angles are under 90 deg and the triangle is acute. Right triangles place the orthocenter at the right-angle vertex; obtuse triangles have their orthocenter outside the triangle; acute triangles contain the orthocenter inside.
Frequently Asked Questions
What is the orthocenter of a triangle?
The orthocenter of a triangle is the point where all three altitudes of the triangle intersect. An altitude is a line segment drawn from a vertex perpendicular to the opposite side. In any triangle, the three altitudes are always concurrent, meaning they meet at a single point called the orthocenter.
How do you calculate the orthocenter from coordinates?
To calculate the orthocenter from vertex coordinates A(x1,y1), B(x2,y2), C(x3,y3): first find the slope of one side (e.g. AB), then compute the perpendicular slope (-1/slope) to get the altitude from the opposite vertex C. Repeat for another side (e.g. AC) to get a second altitude from vertex B. Solve the two altitude line equations as a system of linear equations -- the intersection point is the orthocenter.
Where is the orthocenter located in different triangle types?
In an acute triangle, the orthocenter lies inside the triangle. In a right triangle, the orthocenter coincides with the vertex at the right angle. In an obtuse triangle, the orthocenter lies outside the triangle, beyond the obtuse-angled vertex.
Does every triangle have an orthocenter?
Yes, every triangle has an orthocenter. Even though the three altitudes of a triangle are drawn from each vertex to the opposite side, they always intersect at a single point -- the orthocenter. This holds true for acute, right, and obtuse triangles alike.
Is the orthocenter the same as the circumcenter?
No, in general the orthocenter and circumcenter are different points. The orthocenter is where the three altitudes intersect, while the circumcenter is where the perpendicular bisectors of the sides intersect. Only in an equilateral triangle do the orthocenter, circumcenter, centroid, and incenter all coincide.
How do I find the orthocenter of a right triangle?
In a right triangle, the orthocenter is simply located at the vertex where the right angle is. This is because the two legs of the triangle are themselves altitudes to each other. For example, in a 3-4-5 right triangle, the orthocenter is at the vertex joining the sides of lengths 3 and 4.
What is the orthocenter of an equilateral triangle?
In an equilateral triangle, the orthocenter coincides with the centroid and circumcenter at the same point. Since all sides and angles are equal, all four triangle centers -- orthocenter, centroid, circumcenter, and incenter -- are located at the same interior point.
Is this orthocenter calculator free to use?
Yes, this orthocenter calculator is completely free to use with no registration or download required. All geometry calculators on Calculy are free online tools available to students, teachers, engineers, and professionals worldwide.