## orthocenter Chapter 6 (p. 311)

### The orthocenter is a key point in a triangle that is defined as the point of intersection of the altitudes of the triangle

The orthocenter is a key point in a triangle that is defined as the point of intersection of the altitudes of the triangle. An altitude of a triangle is a line segment drawn from a vertex perpendicular to the opposite side.

To better understand the concept of an orthocenter, let’s consider a triangle ABC. An altitude AH is a line segment drawn from vertex A perpendicular to side BC. Similarly, we can draw altitudes BH and CH from vertices B and C, respectively.

The orthocenter of the triangle ABC is the point of intersection of these three altitudes. It is denoted by the letter H. The orthocenter can be inside the triangle, outside the triangle, or even coincide with one of the vertices of the triangle, depending on the shape and orientation of the triangle.

The orthocenter plays a significant role in geometry and has several notable properties:

1. The orthocenter is always located inside an acute-angled triangle.

2. In a right-angled triangle, the orthocenter is at the vertex of the right angle.

3. In an obtuse-angled triangle, the orthocenter lies outside the triangle.

4. The orthocenter is the center of the nine-point circle, which passes through the midpoints of the sides, the feet of the altitudes, and the midpoints of the line segments connecting the orthocenter to the vertices of the triangle.

5. If a triangle is equilateral, then the orthocenter coincides with its circumcenter and centroid.

Finding the coordinates of the orthocenter can be a bit more complicated, especially in non-standard triangles. However, there are several methods and formulas available, such as using the slopes of the sides, intersecting altitudes, or the circumcenter and incenter of the triangle.

In conclusion, the orthocenter is a significant point in a triangle that is the intersection of the altitudes. It possesses unique properties and is useful in various geometric applications and calculations.

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