Draw the perpendicular bisectors, medians, altitudes and angle bisectors on the triangle. Then look for the concurrency point.
Point A: Circumcenter Point B: Centroid Point C: Incenter Point D: Orthocenter
Practice makes perfect
Let's draw the perpendicular bisectors, medians, altitudes and angle bisectors on the triangle in order to see at which point they intersect.
At first glance, since the centroid and the incenter are always inside the triangle, we can say that neither A nor D can be either the centroid or the incenter.
Finding the Centroid
The centroid is the intersection point of the three medians. Let's draw the medians on the given triangle.
We can see that the medians intersect at point B. Hence, B is the centroid of the triangle.
Finding the Incenter
The incenter is the intersection point of the three angle bisectors. As we've already mentioned, the incenter cannot be A or D and we found the B is the centroid. Therefore, we can conclude that the incenter is point C. However, let's draw the angle bisectors to check that our claim is correct.
Hence, C is the incenter because it is the intersection point of the three angle bisectors, which confirms our claim. Yay!
Finding the Orthocenter
The orthocenter is the intersection point of the three altitudes. Let's draw them on the given triangle.
D is the intersection point of the three altitudes. Consequently, D is the orthocenter.
Finding the Circumcenter
The circumcenter is the intersection point of the three perpendicular bisectors. At this point, by the process of elimination, we know that the circumcenter is point A. However, let's draw the perpendicular bisectors just to verify.
Now we can confidently say that A is the circumcenter because it is the concurrency point of the perpendicular bisectors.
