Enter Triangle Coordinates
Use any consistent coordinate scale. The calculator accepts decimals, negative values, and integer inputs.
Example Data Table
| Vertex A | Vertex B | Vertex C | Orthocenter | Triangle Type | Area |
|---|---|---|---|---|---|
| (0, 0) | (6, 0) | (2, 5) | (2, 1.6) | Scalene, Acute | 15 |
| (1, 1) | (7, 1) | (3, 6) | (3, 2.6) | Scalene, Acute | 15 |
These sample rows show how coordinate input changes the orthocenter and the related geometric properties.
Formula Used
The orthocenter is the intersection point of the triangle’s three altitudes. Each altitude passes through a vertex and is perpendicular to the opposite side.
For side BC, use the direction vector (x₃ - x₂, y₃ - y₂). The altitude through A becomes:
(x₃ - x₂)(x - x₁) + (y₃ - y₂)(y - y₁) = 0
For side AC, the altitude through B becomes:
(x₃ - x₁)(x - x₂) + (y₃ - y₁)(y - y₂) = 0
Solving these two linear equations gives the orthocenter coordinates (Hₓ, Hᵧ). The calculator also computes side lengths with the distance formula and area with the coordinate area formula.
How to Use This Calculator
- Enter the x and y coordinates for vertices A, B, and C.
- Choose the decimal precision for displayed results.
- Click Calculate Orthocenter to solve the triangle.
- Review the orthocenter, altitude equations, side lengths, area, and triangle classification.
- Use the CSV or PDF buttons to save the result summary.
Frequently Asked Questions
1. What is an orthocenter?
The orthocenter is the common intersection point of a triangle’s three altitudes. Each altitude starts at a vertex and meets the opposite side at a right angle.
2. Can the orthocenter lie outside the triangle?
Yes. In an obtuse triangle, the orthocenter lies outside. In an acute triangle, it lies inside. In a right triangle, it sits exactly at the right-angle vertex.
3. Why does the calculator reject collinear points?
Collinear points do not create a valid triangle. Without a real triangle, the altitudes do not form a unique orthocenter, so the calculation must stop.
4. Does the tool support decimal coordinates?
Yes. You can enter integers, decimals, and negative values. The calculator solves the altitude equations using floating-point arithmetic and then formats the output to your selected precision.
5. How are vertical and horizontal sides handled?
The calculator uses line equations in standard form instead of relying only on slopes. That avoids errors when a side or altitude is vertical or horizontal.
6. Why are side lengths and area included?
They provide extra geometric context. Side lengths help classify the triangle, while area confirms the triangle is valid and supports deeper coordinate geometry analysis.
7. Can I use this for homework checking?
Yes. It is useful for verifying manual solutions, checking altitude equations, and reviewing whether the orthocenter should fall inside, outside, or at a vertex.
8. What do the CSV and PDF files contain?
They contain the entered vertices, orthocenter coordinates, triangle type, orthocenter position, side lengths, area, perimeter, and the three altitude equations.