Triangles and TrigonometryProperties of Triangles

Let’s start simple: a triangle is a closed shape that has three sides (which are line segments

) and three vertices (the points where the sides meet). It also has three internal angles, and we already know that the sum of them is always °.

We can classify triangles by the size of their angles:

A right-angled triangle
has one right angle

An obtuse triangle
has one obtuse angle

An acute triangle
has acute angles

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.

Medians

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 ???

. 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 ???

as long as the distance from the centroid to the midpoint.

The centroid is also the “balancing point” of a triangle. Draw a triangle on some cardboard, cut it out, and find the three medians. If you were accurate, you can now balance the triangle on the tip of a pencil, or hang it perfectly level from a piece of string that’s attached to its centroid:

This works because the weight of the triangle is evenly distributed around the centroid. In physics, this point is often called the center of mass.

Perpendicular Bisectors and Circumcircle

Recall that the perpendicular bisector

of a line is the perpendicular line that goes through its ???.

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 ???

The intersection point lies on all three perpendicular bisectors, so it must have the same distance from all three ???

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.

In fact, this means that if you are given any three points, you can use the circumcenter to find a circle that goes through all three of them. (Unless the points are ???

, in which case they all lie on a straight line.)

Angle Bisectors and Incircle

You’re probably getting the hang of this now: we pick a certain construction, do it three times for all sides/angles of the triangles, and then we work out what’s special about their intersection.

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!

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 ???

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

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.

Area and Altitudes

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.

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.

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 ???

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

To calculate the area of a triangle, you can pick any of the three sides as base, and then find the corresponding height, which is the line that is ???

to the base and goes through the opposite vertex.

In triangles, these heights are often called altitudes

. Every triangle has altitudes.

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 ??? the triangle.

In obtuse triangles

, the orthocenter ??? the triangle.

In right-angled triangles

, the orthocenter ??? the triangle. Two of the altitudes are actually just sides of the triangle.