What is the orthocenter?

How triangle type changes its location

In an acute triangle the orthocenter lies inside. In a right triangle it sits at the right-angle vertex because two of the triangle's sides already act as altitudes. In an obtuse triangle it lies outside the triangle.

This center is useful because it connects perpendicular structure, altitude construction, and concurrency in one topic. It also contrasts nicely with the centroid and incenter, which always stay inside the triangle.

The orthocenter is found from altitudes, not from equal distances to vertices or sides. Keeping that source line in mind prevents it from being confused with the circumcenter or incenter when several special points appear in one triangle.

In an acute triangle, three altitude segments meet inside. In an obtuse triangle, their supporting lines meet outside. The word altitude still means the same thing in both cases; only the location of the intersection changes as the triangle changes type.

In a right triangle, the orthocenter is the right-angle vertex because the two legs already serve as perpendicular altitudes. This is a useful limiting case: the concurrency point does not disappear, it simply lands exactly at an existing vertex.

• The orthocenter is the point where the altitudes intersect.
• The orthocenter is formed by the three altitudes.
• Its location depends on triangle type: inside acute, at the right-angle vertex in right triangles, outside obtuse.
• Finding the orthocenter may require extending sides so the altitudes can intersect.

Where the orthocenter helps

• Use the orthocenter when studying altitudes, concurrency, and Euler-line ideas.
• Use it in proofs or constructions that depend on perpendicular segments from vertices.
• Use it to compare how triangle centers behave in different angle classifications.

Three orthocenter checks

• Do not forget that an altitude may hit the extension of a side, not only the side itself.
• Do not assume the orthocenter must be inside the triangle.
• Do not confuse the orthocenter with the circumcenter just because both may lie outside an obtuse triangle.