Bringing it all together
Review of triangle properties
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.