## What is an Orthocenter Calculator?

An Orthocenter Calculator helps you find the orthocenter of a triangle defined by its vertices in a coordinate system. The orthocenter is the point where the three altitudes of a triangle intersect, providing insight into the triangle's geometric properties.

## Formula for Orthocenter Calculation

To find the orthocenter of triangle ABC with vertices A(x₁, y₁), B(x₂, y₂), C(x₃, y₃):

1. Find slopes of the sides.

2. Calculate altitudes.

3. Solve the altitude equations to find the orthocenter coordinates (H).

## How to Use the Orthocenter Calculator

Input the coordinates of the three vertices (A, B, C) in their respective boxes. Click on "Calculate" to find the orthocenter. The result will be displayed in a table format, along with a detailed explanation of the steps taken to solve the problem.

## Result:

Orthocenter H(x, y) |
---|

### Steps to Solve:

### FAQ

#### 1. What is an orthocenter?

The orthocenter is the intersection point of the three altitudes of a triangle. Each altitude is a line segment from a vertex perpendicular to the opposite side.

#### 2. How do I find the orthocenter?

To find the orthocenter, calculate the slopes of the sides of the triangle, determine the equations of the altitudes, and solve for their intersection point.

#### 3. What if my triangle is right-angled?

In a right triangle, the orthocenter is located at the vertex of the right angle, as the altitudes from the other two vertices intersect there.

#### 4. Is the orthocenter always inside the triangle?

The orthocenter lies inside the triangle for acute triangles, on the triangle for right triangles, and outside for obtuse triangles.

#### 5. Can I calculate the orthocenter using any three points?

Yes, as long as the points do not lie on a straight line, they can form a triangle, and you can find the orthocenter.

#### 6. What is the significance of the orthocenter?

The orthocenter is significant in triangle geometry as it helps determine various properties and relations of triangles, especially in relation to circumcircles and incircles.

#### 7. Can the orthocenter be outside the triangle?

Yes, in obtuse triangles, the orthocenter lies outside the triangle, which differentiates it from acute and right triangles.

#### 8. How does the orthocenter relate to the circumcenter?

The circumcenter is the center of the circumcircle and is different from the orthocenter, which relates to the triangle's altitudes. Their positions depend on the type of triangle.

#### 9. How is the orthocenter calculated in a 3D triangle?

In 3D, the concept extends, and the orthocenter is determined using similar principles, though it requires handling additional coordinates and calculations.

#### 10. Can software help in finding the orthocenter?

Yes, various mathematical software and online tools can assist in calculating the orthocenter quickly and accurately without manual calculations.