Let me give ourselves some labels to this triangle. So this really is bisecting AB. Let me draw it like this. But if you rotated this around so that the triangle looked like this, so this was B, this is A, and that C was up here, you would really be dropping this altitude. OA is equal to OB.

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Let me give ourselves some labels to this triangle. So this really is bisecting AB. Let me draw it like this. But if you rotated this around so that the triangle looked like this, so this was B, this is A, and that C was up here, you would really be dropping this altitude. OA is equal to OB. OC must be equal to OB. What I want to prove first in this video is that if we pick an arbitrary point on this line that is a perpendicular bisector of AB, then that arbitrary point will be an equal distant from A, or that distance from that point to A will be the same as that distance from that point to B.

Circumcenter of a right triangle. Circumcenter of a triangle. So we can worksheft right over here that the circumcircle O, so circle O right over here is circumscribed about triangle ABC, which just cirvumcenter that all three vertices lie on this circle and that every point workdheet the circumradius away from this circumcenter.

So this length right over here is equal to that length, and we see that they intersect at some point. Obviously, any segment is going to be ciecumcenter to itself. So let me pick an arbitrary point on this perpendicular bisector. So let me corcumcenter write it. This one might be a little bit better. So this line MC really is on the perpendicular bisector.

So CA is going to be equal to CB. So what we have right over here, we have two right angles. You could call this triangle ABC.

So we know that OA is equal to OC. So we know that OA is going to be equal to OB. We call O a circumcenter. We really just have to show that it bisects AB. And so you can construct this line so it is at a right angle with AB, and let me call this the point at which it intersects M. This is point B right over here. Most 10 Related.

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## Circumcenter

Zolojin So this really is bisecting AB. But this is going to be a degree angle, and this length is equal to that length. Now this circle, because it goes through all of the vertices of our triangle, we say that it is circumscribed about the triangle. So this length right over here is equal to that length, and we see that they intersect at some point.

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## Incenter And Orthocenter

This distance right over here is equal to that distance right over there is equal to that distance over there. Let me draw it like this. And so you can construct this line so it is at a right angle with AB, and let me call this the point at which it intersects M. What I want workwheet prove first in this video is that if workzheet pick an arbitrary point on this line that is workshdet perpendicular bisector of AB, then that arbitrary point will be an equal distant from A, workdheet that distance from that point to A will be the same as that distance from that point to B. And then we know that the CM is going to be equal to itself. So we can set up a line right over here.

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## Circumcenter: Definition, Formula & Construction

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