Triangle Orthocenter Calculator
Find the orthocenter (H) of any triangle from three vertex coordinates. Shows altitude equations, foot of each altitude, and full step-by-step algebraic working.
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| Altitude | From | Perpendicular to | Equation | Foot (on side) |
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Frequently Asked Questions
The orthocenter (H) is the point where the three altitudes of a triangle intersect. An altitude is a line segment from a vertex perpendicular to (or the extension of) the opposite side. For acute triangles, H lies inside; for right triangles, H is at the right-angle vertex; for obtuse triangles, H lies outside.
Find the equations of any two altitudes and solve the system. Altitude from A to BC: slope of BC = (y_C − y_B)/(x_C − x_B); altitude slope = −1/slope_BC; altitude equation: y − y_A = (altitude slope)(x − x_A). Repeat for altitude from B. Solve the two line equations simultaneously to find H.
Acute triangle: H lies inside. Right triangle: H coincides with the right-angle vertex. Obtuse triangle: H lies outside beyond the obtuse vertex. Equilateral triangle: H, centroid, circumcenter, and incenter all coincide at the same point.
The Euler line passes through the orthocenter (H), centroid (G), and circumcenter (O). A remarkable property: G always divides HO in the ratio 2:1 from H. This means OG:GH = 1:2 for any non-equilateral triangle.
Computing the Orthocenter
The orthocenter is found by solving a system of two altitude equations. Each altitude is perpendicular to one side and passes through the opposite vertex. If side BC has slope m, the altitude from A has slope −1/m and passes through A(x_A, y_A): y − y_A = (−1/m)(x − x_A).
Special case: if a side is vertical, its altitude is horizontal (slope 0). If a side is horizontal, its altitude is vertical (undefined slope, written as x = constant).
Triangle Centers Summary
- Centroid G: intersection of medians = ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3)
- Orthocenter H: intersection of altitudes (this calculator)
- Circumcenter O: equidistant from all 3 vertices
- Incenter I: equidistant from all 3 sides
- Euler line: O, G, H are collinear with OG:GH = 1:2