How to Use Orthocenter Calculator
The Orthocenter Calculator finds the orthocenter of any triangle defined by three coordinate points.
- Enter vertex A — type the x and y coordinates of the first vertex into the Orthocenter Calculator.
- Enter vertex B — type the x and y coordinates of the second vertex.
- Enter vertex C — type the x and y coordinates of the third vertex.
- Adjust decimal places — choose how many decimal places you want in the output.
The Orthocenter Calculator instantly shows the orthocenter coordinates H, the equations of two altitudes, and classifies the triangle as acute, right, or obtuse.
Formula & Theory — Orthocenter Calculator
The Orthocenter Calculator uses the altitude intersection method.
Given triangle vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):
Step 1 — Altitude from A perpendicular to BC
The slope of BC is m_BC = (y₃ − y₂) / (x₃ − x₂). The altitude from A has slope m_A = −1 / m_BC and passes through A:
y − y₁ = m_A (x − x₁)
Step 2 — Altitude from B perpendicular to AC
Similarly, the altitude from B has slope m_B = −1 / m_AC and passes through B.
Step 3 — Solve for the intersection H
Setting the two altitude equations equal gives the orthocenter H(x_H, y_H). The Orthocenter Calculator handles all edge cases including vertical sides (undefined slope) automatically.
| Special Case | Behaviour |
|---|---|
| Right triangle | H = right-angle vertex |
| Obtuse triangle | H lies outside the triangle |
| Collinear points | No triangle; Orthocenter Calculator shows error |
Triangle Type Detection
The Orthocenter Calculator classifies the triangle by testing the dot products at each vertex. A negative dot product at any vertex indicates an obtuse angle; a zero dot product indicates a right angle.
Use Cases for Orthocenter Calculator
The Orthocenter Calculator is valuable in a range of academic and professional contexts:
- Geometry courses — students can verify hand calculations or explore how the orthocenter moves as vertices are adjusted using the Orthocenter Calculator.
- Triangle centres exploration — together with the centroid, circumcenter, and incenter, the orthocenter completes the study of triangle centers; the Orthocenter Calculator makes this accessible.
- Computer graphics — orthocenter coordinates are used in barycentric coordinate systems; the Orthocenter Calculator speeds up prototyping.
- Architecture & structural analysis — triangular frameworks require precise geometric analysis; the Orthocenter Calculator helps locate critical structural points.
- Competitive mathematics — the Orthocenter Calculator lets students quickly check answers during problem-solving practice.
Whether you are studying geometry or solving a real-world spatial problem, the Orthocenter Calculator provides accurate, instant results with full step-by-step transparency.