Orthocenter Calculator
Calculate the orthocenter of a triangle given its three vertex coordinates. The orthocenter is the point where the three altitudes of the triangle meet.
Understanding the Orthocenter
The orthocenter is one of the four classical triangle centers and arguably the most geometrically interesting. It is defined as the intersection of the three altitudes, where each altitude is a line from a vertex perpendicular to the opposite side. The fact that all three altitudes are concurrent (meet at one point) is a nontrivial theorem that requires proof, unlike the centroid where concurrency follows directly from the median construction.
What makes the orthocenter distinctive is its variable position relative to the triangle. For acute triangles, it lies inside. For right triangles, it sits exactly at the right-angle vertex. For obtuse triangles, it falls outside the triangle entirely, on the side opposite the obtuse angle. This behavior contrasts with the centroid, which always stays inside, and makes the orthocenter a rich subject for geometric exploration and proof exercises.
Computing the Orthocenter
To find the orthocenter analytically, compute the equations of two altitudes and solve for their intersection. For an altitude from vertex A perpendicular to side BC, the direction of BC is (x_C - x_B, y_C - y_B) and the perpendicular direction is (y_C - y_B, -(x_C - x_B)). The altitude line passes through A in this perpendicular direction.
Repeating this for a second altitude from vertex B perpendicular to side AC gives a second line equation. Solving the two-equation system yields the orthocenter coordinates. This calculator performs this computation for you, handling all cases including when sides are horizontal or vertical. The calculation is more involved than finding the centroid but reinforces important skills in analytic geometry.
The Euler Line and Nine-Point Circle
The orthocenter participates in two beautiful geometric relationships. First, it lies on the Euler line along with the centroid and circumcenter. The centroid divides the segment from the orthocenter to the circumcenter in a precise 2:1 ratio. This alignment holds for all non-equilateral triangles and reveals a deep structural connection between these triangle centers.
Second, the orthocenter relates to the nine-point circle, which passes through the midpoints of the sides, the feet of the altitudes, and the midpoints of the segments from the vertices to the orthocenter. The nine-point circle has its center on the Euler line midway between the orthocenter and circumcenter, with a radius equal to half the circumradius. These elegant relationships demonstrate the rich geometry hidden within every triangle.
Frequently Asked Questions
What is the orthocenter?
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 (or its extension).
Does the orthocenter always lie inside the triangle?
No. The orthocenter lies inside acute triangles, on the hypotenuse vertex for right triangles, and outside obtuse triangles.
How do you find the orthocenter?
Find the equations of at least two altitudes (perpendicular from a vertex to the opposite side) and solve the system of equations to find their intersection point.
What is the Euler line?
The Euler line passes through the orthocenter, centroid, and circumcenter of a triangle. The centroid divides the segment from orthocenter to circumcenter in a 2:1 ratio.
What happens to the orthocenter in a right triangle?
In a right triangle, the orthocenter coincides with the vertex at the right angle, since the two legs are already altitudes.