Orthocenter Calculator
Find the orthocenter of any triangle using vertex coordinates
Triangle Vertices
Vertex A
Vertex B
Vertex C
Load Example Triangles
Orthocenter Results
Orthocenter Coordinates
Triangle Properties
Side Lengths
Altitude Equations
Orthocenter Properties
• The orthocenter lies inside the triangle
Step-by-Step Solution
Given Triangle Vertices
A = (1, 1)
B = (3, 5)
C = (7, 2)
Step 1: Calculate Side Slopes
Slope of BC = (y₃ - y₂) / (x₃ - x₂) = (2 - 5) / (7 - 3) = -0.750
Slope of AC = (y₃ - y₁) / (x₃ - x₁) = (2 - 1) / (7 - 1) = 0.167
Step 2: Calculate Perpendicular Slopes
Perpendicular slope to BC = -1 / slope(BC) = 1.333
Perpendicular slope to AC = -1 / slope(AC) = -6.000
Step 3: Find Altitude Equations
Altitude from A: y = -0.333 + 1.333x
Altitude from B: y = 23.000 + -6.000x
Step 4: Solve for Intersection
The orthocenter is at the intersection of these altitudes:
H = (3.181818, 3.909091)
Triangle Centers
Orthocenter
Intersection of altitudes
Centroid
Intersection of medians
Circumcenter
Intersection of perpendicular bisectors
Incenter
Intersection of angle bisectors
Orthocenter Properties
Altitude Intersection
Where all three altitudes meet
Position Varies
Inside for acute, outside for obtuse
Orthocentric System
Any 3 of 4 points form orthocenter of 4th
Euler Line
Lies on line with centroid and circumcenter
Understanding the Orthocenter
What is the Orthocenter?
The orthocenter of a triangle is the point where all three altitudes intersect. An altitude is a line segment from a vertex perpendicular to the opposite side (or its extension).
Key Properties
- •Acute triangles: Orthocenter lies inside
- •Right triangles: Orthocenter at right-angle vertex
- •Obtuse triangles: Orthocenter lies outside
- •Equilateral triangles: Coincides with other centers
Calculation Method
Step 1: Find slopes of two sides
Step 2: Calculate perpendicular slopes
Step 3: Write altitude equations
Step 4: Solve system of equations
Formulas
Slope: m = (y₂ - y₁) / (x₂ - x₁)
Perpendicular slope: m⊥ = -1/m
Line equation: y - y₁ = m(x - x₁)
Orthocentric System
An interesting property is that any four points form an orthocentric system - if you take any three of them to form a triangle, the fourth point will be the orthocenter of that triangle.
Euler Line
In non-equilateral triangles, the orthocenter lies on the Euler line along with the centroid and circumcenter, with the centroid dividing the segment between orthocenter and circumcenter in a 2:1 ratio.