# Review of triangle properties | Special properties and parts of triangles | Geometry | Khan Academy

What I want to do

in this video is review all of the neat

and bizarre things that we’ve learned

about triangles. So first, we learned–

so let me just draw a bunch of triangles

for ourselves. So let’s have a triangle

right over there. The first thing

that we talked about is the perpendicular bisectors

of the sides of the triangles. So if we take, so let’s bisect

this side right over here. And let’s draw a

perpendicular line to it. So this line right over here

would be the perpendicular bisector of this

side right over here. So it’s bisecting and

it is perpendicular. Let’s draw another perpendicular

bisector right over here. So we’re going to– this is

the midpoint of that side. Let’s draw a perpendicular. So this is perpendicular

and this length is equal to this length, and

then let’s do one over here. This looks like the midpoint

of that side right over there. And then we will

draw a perpendicular. It’s perpendicular, and

we know that this length is equal to this

length right over here. And what we learned is where

all of these perpendicular bisectors intersect, and

what’s neat about this, and frankly all the things

that we’re going to talk about in this video, is that they do

intersect in one unique point. That one unique point is

equidistant from the vertices of this triangle. So this distance, is going to

be equal to this distance, which is going to be equal

to that distance. And because it’s

equidistant to the vertices, you could draw a circle

of that radius that goes through the vertices. And that’s why we call

this right over here, that point, that intersection

of the perpendicular bisectors. So let me write

this down, just so that we can keep

track of things. Perpendicular bisectors. We call this point right

over here our circumcenter, because it is the center of

our circumcircle, a circle that can be circumscribed

about this triangle. So this is our circumcircle. And the radius of the

circumcircle, the distance between the circumcenter and the

vertices is the circumradius. So that was the

perpendicular bisectors. Now the next thing we

learned, and the whole point of this video is

just to make sure that we can differentiate

between these things and not get too confused. So let me draw another arbitrary

triangle right over here. The next thing we

thought about is well, what about if we were

to bisect the angles? So we’re not talking

about perpendicular bisecting the

sides, but we’re now talking about bisecting

the angles themselves. So we could bisect this

angle right over here. Let me draw my best

attempt to draw it. And so this angle is going

to be equal to that angle. We could bisect this

angle right over here. We could bisect– I could do

a better version than that. So that looks–

well, one more try. So I could bisect it like that. And them if I’m

bisecting it this angle is going to be

equal to that angle. And then if I

bisect this one, we know that this

angle is going to be equal to that angle over there. And once again, we have

proven to ourselves that they all intersect

in a unique point. And this point, instead of being

equidistant from the vertices, this point is equidistant from

the sides of the triangle. So if you dropped

a perpendicular to each of the sides. So this distance is going to be

equal to that distance, which is going to be equal

to that distance. And because of that,

we can draw a circle that is tangent to the

sides that has this radius. So we could draw a circle

that looks like this. And we call this

circle, because it’s kind of inside the triangle,

we call it an incircle. And this point,

we can call, which is the intersection of

these angle bisectors, we can call this the inradius. Now the other thing we

learned about angle bisectors, and this we just

have to draw one. So let me just draw another

triangle right over here. And let me draw

an angle bisector. So I’m going to

bisect this angle. So this angle is

equal to that angle. And let me label

some points here. So let’s say that this

is– change the colors. Let’s say that is A,

this is B, this is C, and this is D. We learned

that if AC is really the angle bisector of angle BAD, that the

ratio between– that AB over BC is going to be equal to

the ratio of AD to DC. Sometimes this is called

the angle bisector theorem. So that’s neat. So the next thing we

learned is– let’s draw another triangle here. This is just to be an

overview of everything we’ve been covering in

the last few videos. So let me draw

another triangle here. So now instead of drawing the

perpendicular bisector– so let me label everything. This was angle bisectors. And now what I’m going to

think about are the medians. So the perpendicular bisectors

were from the midpoint, were lines that

bisect the sides, and they are perpendicular,

but don’t necessarily go through the vertices. When we talk about

medians, we are talking about points

that bisect the sides, but they go to the

vertices, and they’re not necessarily perpendicular. So let’s draw some medians here. So let’s say this is the

midpoint of that side right over there. So we could draw a

median like that. No, this is going

through the vertices, these did not necessarily

go through the vertices. This right over here is not

necessarily perpendicular. But we do know

that this length is equal to that length

right over there. Let me draw a couple of more

medians right over here. So this, the midpoint looks

like it’s right about here. So this length is

equal to that length, and notice it goes

through the vertex, but it’s not necessarily

perpendicular. And then this one–

see the midpoint looks like it’s

right about there. And once again, all of

these are concurrent. They all intersect at one

point right over here. And so this length

right over here is equal to this

length right over here. There’s a bunch of neat

things about medians. When you draw the three medians

like this, that unique point where they intersect, we

called it the centroid. And as I mentioned, and

you might learn this later on in physics, is if

this was a uniform triangle, if it had a uniform

density, and if you were to throw it or

rotate it in the air, it would rotate around

its centroid, which would essentially be

its center of mass. It would rotate around that as

it’s flying through the air. If it had some type of

rotational, or I guess you could say angular momentum. But the neat thing about this

is it also divides this triangle into six triangles

of equal area. So this triangle has the

same area as that triangle. We proved this in

several videos ago. Each of these six triangles

all have the same area. The other thing that we

learned about medians is that where the centroid

sits on each of the medians is 2/3 along the median. So the ratio of this side, of

this length to this length, is 2 to 1. Or this is 2/3 along

the way of the median. This is 2/3 of the median,

this is 1/3 of the median. So the ratio is 2 to 1. Another related

thing we learned, this wasn’t really

necessarily about medians, but it’s a related concept, was

the idea of a medial triangle. A medial triangle

like this, where you take the midpoint

of each side, and you draw a

triangle that connects the midpoints of each side. We call this triangle

a medial triangle. And we proved to

ourselves that when you draw a medial triangle,

it separates this triangle into four triangles that

not only have equal area, but the four triangles here are

actually congruent triangles. And not only are they

congruent, but we’ve shown that this side is

parallel to this side. That– let me use some

more colors here– this side is parallel. Actually, I shouldn’t

draw two arrows like that. That side is parallel

to that side. This side is parallel

to this side. And then you have this side

is parallel to this side right here. And this length is

1/2 of that length, this length is 1/2

of that length, this length is 1/2

of that length. And it really just

comes out of the fact that these are four

congruent triangles. And then the last thing

that we touched on is drawing altitudes

of a triangle. So there’s medians,

medial triangles, and I’ll draw one last

triangle over here. And here, I’m going to go

from each of the vertex, and I’m not going to go to the

midpoint of the other side. I’m going to drop

a perpendicular to the other side. So here I will drop

a perpendicular, but this isn’t necessarily

bisecting the other side. Once again, going to

drop a perpendicular but not necessarily

bisecting the other side. And then drop a

perpendicular but not necessarily bisecting

the other side. And we’ve also

proven to ourselves– so these are the

altitudes of the triangle. And these also intersect

in a unique point. And I want to be clear,

this unique point does not necessarily have to

be inside of the triangles. And the same thing was true of

the perpendicular bisectors. It actually could be

outside of the triangle. And this unique point

we call an orthocenter. So I’ll leave you there. And hopefully this

was useful, because I know it can get confusing. How’s a median different

than a circumcenter, which is different than an

orthocenter, or an inradius, or any of these type of things? So hopefully this clarified

things a little bit.