It’s Professor Dave, let’s learn more about triangles.
Let’s quickly identify some other lines that we can draw on triangles.
One line we could draw is a perpendicular bisector.
This is a line that is perpendicular to one of the sides of the triangle and cuts that
side into two line segments of equal length.
Any point on this bisector must be equidistant from the endpoints of these line segments,
so if the bisector also coincides with the other two sides, then these two sides must
be equal, and we have a right triangle.
This also makes the two smaller triangles congruent.
Besides perpendicular bisectors, we can also describe angle bisectors, which are lines
that cut an angle in half.
Any point on this angle bisector will be equidistant from the two other lines.
Now let’s see what happens if we have a triangle, and we try to draw three perpendicular
bisectors, starting from the middle of each side.
These extend from each midpoint at a ninety degree angle, and meet at some location, called
the circumcenter of the triangle.
The circumcenter will be inside the triangle if it is acute, outside of the triangle if
it is obtuse, and on the triangle if it is a right triangle.
While the distance to the sides of the triangle may differ, the circumcenter will be equidistant
from the vertices of the triangle, meaning that the distance from the circumcenter to
each corner is the same, no matter what type of triangle we are looking at.
The circumcenter of the triangle will also be the center of the circle that circumscribes
that triangle.
This is the circle that contains all three vertices of the triangle.
We can also discuss what happens if we bisect all the angles of a triangle.
These angle bisectors will meet somewhere called the incenter of the triangle, and unlike
the circumcenter, the incenter will always be inside the triangle.
The incenter will be equidistant from the sides of the triangle, so these three lengths
must be the same.
The incenter will be the center of the circle that is inscribed in the triangle, which is
the circle that intersects each side of the triangle at precisely one point.
These theorems can be used to prove all kinds of things about various triangles, like the
measure of certain angles and the lengths of certain line segments, we just use the
postulates strategically.
Now instead of a bisector, let’s look at something called a median.
A median of a triangle is a line segment that connects one vertex and the midpoint of the
opposite side.
Notice that this is not a bisector only because these are not right angles, if they were right
angles, this median would also be a bisector, and we would have a right triangle.
But it’s not, so it’s just a median.
If we find the point where all three medians meet, this is called the centroid of the triangle,
and like the incenter, this is always inside the triangle.
The centroid is always two thirds of the way from one vertex to the opposite side, so that
means that this distance is twice this, this distance is twice this, and this distance
is twice this.
This allows us to determine the lengths of certain line segments given others.
We can also identify the altitude of a triangle.
This is what we get when we start from a vertex and make a line segment that extends towards
the opposite side in perpendicular fashion.
If we have to extend the opposite side in one direction or the other to get this, that’s
what we do, we just need to make a right angle.
Each triangle has three altitudes, one per vertex, and the place where these altitudes
converge is called the orthocenter of the triangle.
So to quickly summarize, the point where perpendicular bisectors meet is called the circumcenter.
Notice the right angles at the midpoints of the sides.
This will be equidistant from the vertices.
The point where the angle bisectors meet is called the incenter.
This will be equidistant from the sides.
The point where the three medians meet is called the centroid, and the point where the
three altitudes meet is called the orthocenter.
Again, notice the right angles required to get the orthocenter, whereas for the centroid
the angles can have any value.
Let’s check comprehension.