# The Orthocenter

- Author:
- Jorge Cássio

- Topic:
- Orthocenter, Special Points, Triangles

## Orthocenter

The orthocenter of a triangle is the point of intersection of any two of three altitudes of a triangle (the third altitude must intersect at the same spot). The orthocenter can also be considered as a point of concurrency for the supporting lines of the altitudes of the triangle.

## Drawing (Constructing) the Orthocenter

Let's build the orthocenter of the ABC triangle in the next app.

- The line segment needs to intersect point
**C**and form a right angle (90 degrees) with the "suporting line" of the side AB. Definition of "supporting line: The supporting line of a certain segment is the line - which contains that segment" The first thing to do is to draw the "supporting line". Enable the tool LINE (Window 3) and click on points
**A**and**B.**A line**d**will appear. - Enable the tool PERPENDICULAR LINE (Window 4), click on vertex
**C**and then click on the supporting line**d**or on side c¹.

- Select the tool INTERSECT (Window 2). Click on the lines
**d**and**e**. The point**D**will be appear.

**Note:**The segment CD is called the altitude of the triangle, because it connects a vertex in a perpendicular way (forming a 90 degree angle) to the supporting line of the opposite side (of the vertex). Line**e**is the supporting line to the altitude relative to side**AB**. - The altitude relative to side
**BC**must be drawn now. First, let's draw the supporting line relative to side**BC**. Enable the tool LINE (Window 3) and click on vertices**C**and**B.**A line “**f”**will appear. - Enable the tool PERPENDICULAR LINE (Window 4), click on vertex
**A**and then on the line**“f”**or on side**“a”**. - Enable the tool INTERSECT (Window 2), click on line
**f**and then on line**g**. A vertex**E**will appear. This vertex is the foot of the altitude relative to side**BC**. Line**g**is the supporting line of that altitude. - Now there are two supporting lines to the altitudes, correct? One relative to side
**AB**and the other relative to side**BC**. Shall we mark where these lines intersect? With the tool INTERSECT TWO OBJECTS (Window 2) still enabled, click on line**e**(supporting line to the altitude relative to side**AB**) and on line "**g"**; (supporting line to the altitude relative to side**BC**). A new point will appear (point**F**). This point is the orthocenter of the triangle. - Enable the tool MOVE GRAPHICS VIEW (Window 11) to adjust the position of the objects in
the Viewing Window and use the
*mouse*to adjust the*zoom*. If all the procedures were followed correctly, you will have a figure similar to the one below.

- The altitude relative to side
**BC**must be drawn now. First, let's draw the supporting line relative to side**AC**. Enable the tool LINE (Window 3) and click on vertices**A**and**C.**A line “**h”**will appear. - Enable the tool PERPENDICULAR LINE (Window 4), click on vertex
**B**and then on the line**“h”**or on side**“b”**. A line**i**will appear. Does this line go through the point**F**(orthocenter)? From this moment point**F**will be*called***orthocenter.**In order to do this, right click the*mouse*on vertex**F**and select the option RENAME. In the new window that appears,**type**orthocenter and click OK.

## Analysis 1

When will the triangle have an internal orthocenter? When will the triangle have an external orthocenter? When will the orthocenter coincide with one of the vertices?

## Property

The supporting lines of the altitudes of a triangle intersect at the same point.

## Analysis 2

If the orthocenter would lie outside the triangle, would the theorem proof be the same?

## Angle formed by altitudes

## Analysis 3

Move the vertices of the previous triangle and observe the angle formed by the altitudes. When will this angle be acute? When will this angle be obtuse?