Orthocenter Calculator — Find the Orthocenter of a Triangle
Enter the (x, y) coordinates of all three vertices and the calculator solves for the orthocenter — the point where the three altitudes of the triangle meet — using a system of perpendicularity equations.
Point where all three altitudes of the triangle intersect
- 1
Determinant (a·d − b·c)
-4 × 4 − 4 × 2 = -24Zero determinant means the vertices are collinear — no triangle exists. - 2
Constant e = a·x₁ + b·y₁
-4 × 0 + 4 × 0 = 0 - 3
Constant f = c·x₂ + d·y₂
2 × 6 + 4 × 0 = 12 - 4
Numerator = e·d − b·f
0 × 4 − 4 × 12 = -48 - 5
Orthocenter x = numerator ÷ det
-48 ÷ -24 = 2
How does this calculator work?
The orthocenter H is where the three altitudes of a triangle intersect. For vertices A, B, C, solve the two conditions AH ⊥ BC and BH ⊥ AC simultaneously (a 2×2 linear system). H lies inside an acute triangle, at the right-angle vertex for a right triangle, and outside for an obtuse triangle.
Formula
How this is calculated
The orthocenter H is the common intersection of the three altitudes (lines through each vertex, perpendicular to the opposite side). Because any two altitudes determine H uniquely, it is sufficient to intersect just two of them.
This calculator sets up the perpendicularity conditions algebraically: the vector AH must be perpendicular to BC (their dot product is zero), and BH must be perpendicular to AC. These give two linear equations in the unknowns hx and hy, which are solved by Cramer's rule. If the determinant is near zero — meaning the vertices are collinear — no triangle exists and no result is shown.
For an acute triangle, H lies inside the triangle. For a right triangle, H coincides with the right-angle vertex. For an obtuse triangle, H falls outside the triangle on the far side of the obtuse angle. The plotted diagram marks all three vertices (A, B, C) and the orthocenter (H) so you can see immediately which case applies.
Frequently asked questions
The orthocenter is the intersection of the three altitudes (perpendiculars from each vertex to the opposite side). The centroid is the intersection of the three medians (vertex-to-midpoint lines). The circumcenter is equidistant from all three vertices. All three are collinear on the Euler line for non-equilateral triangles.
The altitude from an obtuse vertex still falls inside, but the altitudes from the two acute vertices must be extended beyond the opposite sides to meet. That extension places H outside the triangle, on the opposite side from the obtuse angle.
In a right triangle the two legs are themselves perpendicular to each other, so the altitude from the right-angle vertex falls along one leg. Both altitudes from the acute vertices pass through the right-angle vertex. The orthocenter is therefore exactly at the right-angle vertex.
Also known as
TG we-Calculate Editorial Team. (2026). Orthocenter Calculator — Find the Orthocenter of a Triangle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/orthocenter-calculator
TG we-Calculate Editorial Team. "Orthocenter Calculator — Find the Orthocenter of a Triangle." TG we-Calculate. 2026. https://we-calculate.com/calculator/orthocenter-calculator.
TG we-Calculate Editorial Team, "Orthocenter Calculator — Find the Orthocenter of a Triangle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/orthocenter-calculator
@misc{wecalculate_orthocenter_calculator, title = {Orthocenter Calculator — Find the Orthocenter of a Triangle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/orthocenter-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?