Free · Mathematics models
Triangle Orthocenter from Coordinates
Intersect two altitude lines to find the orthocenter of a nondegenerate coordinate triangle, including right triangles.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
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Formula and method
For vertices (0,0), (2,0) and (1,2), the orthocenter is (1,0.5).
Solve two perpendicular-altitude dot-product equations; their determinant is the signed double area.
Worked example
Enter these known values and leave the other values blank.
- x₁
- 0
- y₁
- 0
- x₂
- 2
- y₂
- 0
- x₃
- 1
- y₃
- 2
- x
- 1
- y
- 0.5
Assumptions and limitations
- Three finite points must be noncollinear. The altitude intersection can lie inside, outside or at a vertex of the triangle.
- The direct coordinate formula handles vertical sides and right-angle vertices; the original side and angle controls remain preserved. Hidden slope intermediates can be undefined for vertical sides without making the altitude intersection undefined.
- Coincident or collinear vertices do not determine an orthocenter.
- Clear the unknown numerical field before using a supported inverse; selection controls retain their captured branches.
Common questions
What assumptions matter for this model?
Three finite points must be noncollinear. The altitude intersection can lie inside, outside or at a vertex of the triangle.
What does this model leave out?
Coincident or collinear vertices do not determine an orthocenter.
References
Model requirements: Vertices must form a nondegenerate triangle. Authored accessible explanations preserve each captured diagram mode and its geometric relationships. Intersect coordinate altitude lines directly instead of weighting vertices by tan(angle); the captured tangent form is singular at right-angle vertices. Angle targets use coordinate cross/dot atan2 to avoid acos cancellation near collinearity, while all captured side-length inverses remain.