Define the orthocenter of the triangle.

The orthocenter of a triangle is defined as the point where the three altitudes of the triangle intersect.

An altitude of a triangle is a perpendicular line segment drawn from a vertex to the opposite side (or the line containing the opposite side). Since a triangle has three vertices, it also has three altitudes.

Key points about the orthocenter:

The orthocenter lies inside the triangle if the triangle is acute (all angles are less than 90 degrees).
The orthocenter lies outside the triangle if the triangle is obtuse (one angle is greater than 90 degrees).
The orthocenter lies on the vertex of the right angle in a right triangle.
Detailed Explanation
Let’s consider a triangle ABC:

Draw the altitude from vertex A to side BC. This is a line perpendicular to BC passing through A.
Draw the altitude from vertex B to side AC. This is a line perpendicular to AC passing through B.
Draw the altitude from vertex C to side AB. This is a line perpendicular to AB passing through C.
The three altitudes will intersect at a single point, which is called the orthocenter.

Properties
The orthocenter is one of the four main points of concurrency in a triangle, along with the centroid, circumcenter, and incenter.
The position of the orthocenter depends on the type of triangle (acute, obtuse, or right).
By studying the orthocenter, mathematicians explore properties of perpendicularity and concurrency in geometry, which have applications in construction, navigation, and design.