okay let's start with the in-center so when we're talking about the in-center what we're looking at are the three angle bisectors so it's this ray here is cutting this angle in half we draw another angle bisector from this vertex that's cutting this angle in half and if we draw one more from the the third vertex cutting that angle in half What we have is this point right here, which is called the incenter. And the reason they call it the incenter is because you can inscribe the circle inside of the triangle such that the sides of the triangle are tangent to the circle. Now, what does that mean when you say that they're tangent to the circle? It means that they just barely touch at one point.
But if you draw a perpendicular right angle, okay, from that center of that circle to the sides of the triangle like this, these are going to be the radii of the circle. And what do you know about the lengths of the radii of a circle? They're all congruent. So if you can identify that you're looking at the three angle bisectors, this point of concurrency as it's called is referred to as the in-center. Think of inscribing the circle inside and then you're going to know that the three radii are congruent.
Now let's talk about the circumcenter. So when we talk about the circumcenter, what we're talking about here are the three perpendicular bisectors. So when you think of perpendicular, we think of at a right angle, right? When we think of a bisector, we think of it as cutting the side in half or bisecting.
And so if we draw the three perpendicular bisectors, let me see if I can draw this accurately here for us. Okay, what we end up with here, this point where all three of those perpendicular bisectors cross, is called the circumcenter. And the reason they call it the circumcenter is because what you can do is you can circumscribe or draw a circle around the triangle, okay, such that the vertices of the triangle lie right on the circle.
Now what they're going to ask you is something referring to the length of the radii. If you look at this radius here and you look at this radius here and you look at this radius here, all these radii are going to be congruent because the radii in a circle are congruent. Now this isn't a right triangle and so what you notice is that The point of concurrency, that circumcenter, is lying right on the triangle.
Okay, so I'm just going to write over here, on. Let's look at what happens when you have an obtuse triangle like this one here. We draw the three perpendicular bisectors.
So here's a perpendicular bisector. bisector okay perpendicular cuts the side in half let's see here is a perpendicular bisector okay cuts this side in half and let's see where's the third one it would be somewhere like right about here okay and it cuts this side in half now my drawing is just a rough drawing here but what you can understand is that when it's that obtuse triangle like this is that the point of concurrency that circumcenter is going to be outside of the triangle so when you go to draw the circle around the triangle okay that center that circle is going to be outside so we'll just write out for this one and then for a acute triangle where all the angles are less than 90 if we draw the three perpendicular bisectors Okay, what you notice is that point of concurrency, that circumcenter, lies inside the triangle. So I'm just going to write in here.
Now, one thing to pay attention to is that, see, look at this triangle over here. Notice that when I drew the perpendicular bisectors, the perpendicular bisector doesn't necessarily go through the opposite vertex. Now, it happened to in this triangle over here. It also happened over here in this triangle here, but that's not always the case.
You just want to make sure that you go through the midpoint of the side in perpendicular, and then... where those three cross, that's going to be your circumcenter. For the centroid, what we're looking at are the three medians. So where do the three medians cross?
That's the point of concurrency. and we call that point where they cross the centroid. But what you do with the centroid is you go from the vertex to the middle of the opposite side. So the median you think of middle.
So these two are going to be congruent. We're going from the opposite vertex to the opposite side. That's one median.
Let's find the midpoint over here, this side, and draw that median. And then same thing over here, let's find the midpoint of this third side. and draw from the vertex to the opposite side. Where those three points cross, that's our centroid. And the important thing to remember about the centroid is that it's like the center of mass.
If I was to cut this out here, this triangle, and I could balance it right at this point. Okay, that's one key characteristic. The other key characteristic is that where, okay, This point crosses, it divides up the median into two pieces.
It's dividing up into this portion right here, which is one-third of the whole median length, and this other piece right here, which is two-thirds. Now, how do you know which is the one-third or two-thirds piece? Well, usually you can tell because the one-third piece is, you know, obviously shorter, but sometimes it's not so easy to tell. So, when you're going from the vertex towards the opposite side, that's going to be the two-thirds portion.
from the side to the centroid, that's going to be the one-third portion. So here if you see from this vertex, see that's two-thirds, one-third. Now just to make this a little bit more concrete, say that the length of this median was like 27. All we'd have to do is multiply 27 times one-third or divide it by three. We would know that this length is nine.
We could then double it because two-thirds is double one-third and we then know this length is 18. Of course you could multiply 27 by two-thirds as well. So that's the key characteristics of a centroid. Now talking about the orthocenter, okay.
That's where the three altitudes cross, and it depends on the type of triangle that we're looking at here. There's kind of three different things that can occur. When it's a right triangle, see, the altitude is, you know, you can think of a triangle as like a mountain, right?
If you climb up this mountain and somebody says, well, how high above the ground are you? You would measure straight down. That's your altitude, okay? So in this case, you can see the altitude lies right along the edge of this triangle. If I was to take this triangle and rotate it, okay, or if I was looking from this vertex, perpendicular to the opposite side.
This is an altitude. And if I rotate it again, sometimes it's helpful on a sheet of paper just to think of this as a mountain, like I was saying. But what you're doing is you're going from the vertex perpendicular to the opposite side. Now notice all three of these altitudes, they cross right at this vertex where the right angle is. So with the orthocenter, the three altitudes are going to cross, I'm going to say, on the triangle when it's a right triangle.
When it's an acute triangle, the angles are less than 90. Let's draw the altitudes here. So again, going from the vertex perpendicular to the opposite side. Notice they're crossing inside. That's the point of concurrency, the orthocenter.
And when it's an obtuse triangle, this sometimes tricks students a little bit. When you climb up this mountain here and you say,"'Okay, how high above the ground am I?'Well, it's almost like you could think of this as being like water over here, right? So you have to actually extend this opposite side here and you're going perpendicular to that opposite side. Now what you're going to have to do is you're going to have to extend that a little bit further, okay, and same thing here. If we find from the vertex perpendicular to the opposite side, extend that, okay, and then same thing over here. We'd have to go from this vertex perpendicular to the opposite side. We're going to have to extend this side and should, again, you know, this is just a rough drawing. Perpendicular, you can see they're crossing outside when it's an obtuse triangle. So those are the key characteristics. If you want to see how to actually do these constructions with a compass and a straight edge, follow me over to that video right there and we'll talk more about these.