Find the orthocenter of a triangle, where all three altitudes intersect.
Last updated: April 2026 | By Patchworkr Team
The orthocenter of a triangle is the point where all three altitudes intersect. An altitude is a line segment from a vertex perpendicular to the opposite side (or its extension). Unlike the centroid, circumcenter, or incenter, the orthocenter may lie outside the triangle. For acute triangles, it lies inside; for right triangles, it coincides with the vertex of the right angle; for obtuse triangles, it lies outside.
The orthocenter has remarkable properties in Euclidean geometry. The reflection of the orthocenter over a side of the triangle lies on the circumcircle. The four points formed by the orthocenter and the three vertices create an orthocentric system, where each point is the orthocenter of the triangle formed by the other three. These properties make the orthocenter crucial in advanced geometry problems, coordinate geometry, and in understanding the deeper structure of triangles.
Label your triangle's vertices as A, B, and C with coordinates (x₁, y₁), (x₂, y₂), (x₃, y₃).
Calculate the slopes of two sides of the triangle using m = (y₂ − y₁) / (x₂ − x₁).
The slopes of altitudes are negative reciprocals: m⊥ = −1/m. These are perpendicular to the sides.
Using point-slope form, write equations for two altitudes. The orthocenter is their intersection.
Solve the system of two altitude equations to find the x and y coordinates of the orthocenter.
Yes, for obtuse triangles. For acute triangles it's inside; for right triangles it's at the right angle.
The centroid is where medians intersect; the orthocenter is where altitudes intersect. They're different points.
Both are special triangle centers. The line through them is part of the Euler line, a key geometric concept.
Yes, every non-degenerate triangle has exactly one orthocenter (where altitudes meet).
This can't happen in a valid triangle. Altitudes always intersect at exactly one point.
The orthocenter should lie on all three altitudes. You can verify by checking it satisfies all three altitude equations.
The four points formed by the orthocenter and three vertices, where each is the orthocenter of the triangle formed by the other three.
Yes, draw the three altitudes (perpendiculars from vertices to opposite sides). Their intersection is the orthocenter.
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