Triangles and TrigonometryProperties of Triangles

For convenience, we always label triangles in the same way. The vertices are labelled with capital letters A, B and C, the sides are labelled with lowercase letters a, b and c, and the angles are labelled with Greek letters α, β and γ (“alpha”, “beta” and “gamma”).

The side that lies opposite vertex A is labeled a, and the angle that lies right next to A is labelled α. The same pattern works for B/b/β and for C/c/γ.

Here you can see a triangle as well as the midpoints of its three sides.

A median of a triangle is a line segment that joins a vertex and the midpoint of the opposite side. Draw the three medians of this triangle. What happens as you move the vertices of the triangle?

It seems like the medians always intersect in one pointhave the same lengthdivide each other in the middle. This point is called the centroid.

Medians always divide each other in the ratio 2:1. For each of the three medians, the distance from the vertex to the centroid is always twicethree timesexactly as long as the distance from the centroid to the midpoint.

Recall that the perpendicular bisector of a line is the perpendicular line that goes through its midpointendpoints.

Draw the perpendicular bisector of all three sides of this triangle. To draw the perpendicular bisector of a side of the triangle, simply click and drag from one of its endpoints to the other.

Like before, the three perpendicular bisectors meet in a single point. And again, this point has a special property.

Any point on a perpendicular bisector has the same distance from the two endpoints of the lines it bisects. For example, any point on the blue bisector has the same distance from points A and C and any point on the red bisector has the same distance from points A and BA and CB and C.

The intersection point lies on all three perpendicular bisectors, so it must have the same distance from all three verticessides of the triangle.

This means we can draw a circle around it that perfectly touches all the vertices. This circle is called the circumcircle of the triangle, and the center is called the circumcenter.

Recall that the angle bisector divides an angle exactly in the middle. Draw the angle bisector of the three angles in this triangle. To draw an angle bisector, you have to click on three points that form the angle you want to bisect.

Once again, all three lines intersect at one point. You probably expected something like this, but it is important to notice that there is no obvious reason why this should happen – triangles are just very special shapes! Continue

Points that lie on an angle bisector have the same distance from the two lines that form the angle. For example any point on the blue bisector has the same distance from side a and side c, and any point on the red bisector has the same distance from sides a and ba and cb and c.

The intersection point lies on all three bisectors. Therefore it must have the same distance from all three sidesvertices of the triangle.

This means we can draw a circle around it, that lies inside the triangle and just touches its three sides. This circle is called the incircle of the triangle, and the center is called the incenter.

Finding the area of a rectangle is easy: you simply multiply its width by its height. Finding the area of a triangle is a bit less obvious. Let’s start by “trapping” a triangle inside a rectangle. Continue

The width of the rectangle is the length of the bottom side of the triangle (which is called the base). The height of the rectangle is the perpendicular distance from the base to the opposite vertex. Continue

The height divides the triangle into two parts. Notice how the two gaps in the rectangle are exactly as big as the two parts of the triangle. This means that the rectangle is twice asthree times asexactly as large as the triangle.

We can easily work out the area of the rectangle, so the area of the triangle must be half that:

A=12× base × height

Like the medians, perpendicular bisectors and angle bisectors, the three altitudes of a triangle intersect in a single point. This is called the orthocenter of the triangle.

In acute triangles, the orthocenter lies insidelies outsideis a vertex of the triangle.

In obtuse triangles, the orthocenter lies outsidelies insideis a vertex of the triangle.

In right-angled triangles, the orthocenter is a vertex oflies insidelies outside the triangle. Two of the altitudes are actually just sides of the triangle.