Orthocenter Calculator
Calculate orthocenter instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
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)\nAltitude from A perpendicular to BC is vertical: x = 0\nSlope of AC = (0 - 6)/(3 - 0) = -2\nAltitude from B perpendicular to AC has slope = 1/2\nEquation: y - 0 = 0.5(x - (-3)) => y = 0.5x + 1.5\nAt x = 0: y = 1.5\nOrthocenter = (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\nAltitude from A: slope = 4, equation: y = 4x\nSlope of AC = (1 - 0)/(1 - 0) = 1\nAltitude from B: slope = -1, equation: y = -(x - 5) = -x + 5\nIntersection: 4x = -x + 5 => 5x = 5 => x = 1, y = 4\nOrthocenter = (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.
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