Enter the coordinates of your triangle's three vertices — Point A (x₁, y₁), Point B (x₂, y₂), and Point C (x₃, y₃) — and the Triangle Orthocenter Calculator finds the exact orthocenter (H), the point where all three altitudes intersect. You'll see the H(x, y) coordinates along with the slopes of each altitude line.
Orthocenter X Coordinate
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Orthocenter Y Coordinate
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Altitude from A (slope)
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Altitude from B (slope)
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Altitude from C (slope)
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The orthocenter is the point where all three altitudes of a triangle intersect. An altitude is a line segment drawn from a vertex perpendicular to the opposite side (or its extension). Every triangle has exactly one orthocenter, and the three altitudes are always concurrent — they always meet at a single point.
To find the orthocenter, you set up two altitude equations and solve them simultaneously. For each altitude, find the slope of the opposite side, take its negative reciprocal (perpendicular slope), and write the line equation through the vertex. Solving any two of those equations gives you the orthocenter H(x, y).
For an acute triangle, the orthocenter lies inside the triangle. For a right triangle, it coincides exactly with the vertex at the right angle. For an obtuse triangle, the orthocenter falls outside the triangle, beyond the obtuse vertex.
No, they are different points. The circumcenter is equidistant from all three vertices and is the center of the circumscribed circle. The orthocenter is the intersection of the three altitudes. They coincide only in an equilateral triangle, where the centroid, circumcenter, incenter, and orthocenter all fall on the same point.
No. The orthocenter is not generally equidistant from the three vertices — that property belongs to the circumcenter. The orthocenter's position is determined by the altitudes, not by equal distances to the vertices.
For a 3-4-5 right triangle, the orthocenter is located at the vertex of the right angle. Since the two legs of a right triangle are themselves altitudes, they intersect at the right-angle vertex, making that point the orthocenter.
To construct the orthocenter, draw an altitude from each vertex to the opposite side. For each altitude, use your straightedge to extend the opposite side if needed, then use your compass to construct a perpendicular from the vertex to that side. Repeat for two vertices — the intersection of just two altitudes gives the orthocenter, and the third will pass through the same point.
Yes. For obtuse triangles, the orthocenter lies outside the triangle. This happens because the altitudes from the two acute vertices must be extended beyond the triangle's sides to meet the altitude from the obtuse vertex. For acute triangles it is always inside, and for right triangles it is exactly on the boundary at the right-angle vertex.