Orthocenter Calculator
Calculate the orthocenter of a triangle - the intersection of altitudes
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Orthocenter (H)
Intersection of altitudes
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About the Orthocenter
The orthocenter is the point where the three altitudes of a triangle intersect. An altitude is a line segment from a vertex perpendicular to the opposite side (or the line containing the opposite side).
Key Properties
- For an acute triangle, the orthocenter lies inside the triangle
- For a right triangle, the orthocenter is at the vertex of the right angle
- For an obtuse triangle, the orthocenter lies outside the triangle
- The orthocenter, centroid, and circumcenter are collinear (lie on the Euler line)
- Each altitude is perpendicular to one side of the triangle
Calculation Method
The orthocenter is calculated by finding the intersection of two altitudes:
- Find the slope of each side
- Calculate the perpendicular slope (negative reciprocal)
- Form altitude equations through vertices
- Solve for the intersection point
Frequently Asked Questions
What is the orthocenter of a triangle?
The orthocenter is the point where the three altitudes of a triangle intersect. An altitude is a perpendicular line from a vertex to the opposite side, and all three altitudes meet at this single point.
Can the orthocenter be outside the triangle?
Yes, for obtuse triangles, the orthocenter lies outside the triangle. For acute triangles, it's inside, and for right triangles, it's located at the vertex of the right angle.
How do you calculate the orthocenter?
To calculate the orthocenter, find the equations of two altitudes (perpendicular lines from vertices to opposite sides) and solve for their intersection point. The perpendicular slope is the negative reciprocal of the side's slope.
What is the Euler line?
The Euler line is a straight line that passes through the orthocenter, centroid, and circumcenter of any triangle (except equilateral triangles where these points coincide). The centroid divides the segment between orthocenter and circumcenter in a 2:1 ratio.
What's special about the orthocenter of a right triangle?
In a right triangle, the orthocenter is located exactly at the vertex where the right angle is formed. This is because the two legs of the right triangle are themselves altitudes.
Why is the orthocenter important?
The orthocenter is one of the four classical triangle centers and is important in geometry, computer graphics, and engineering. It's used in structural analysis, geometric constructions, and understanding triangle properties.
What's the difference between altitude and median?
An altitude is perpendicular to a side and passes through the opposite vertex, intersecting at the orthocenter. A median connects a vertex to the midpoint of the opposite side and intersects at the centroid. These are different lines with different properties.
Does every triangle have an orthocenter?
Yes, every triangle has an orthocenter. However, its location varies: inside for acute triangles, at a vertex for right triangles, and outside for obtuse triangles. Three non-collinear points always define a unique orthocenter.