Triangle Centers

Where is the center of a triangle?
There are actually thousands of centers!
Here are the 4 most popular ones:

Centroid, Circumcenter, Incenter and Orthocenter

For each of those, the "center" is where special lines cross, so it all depends on those lines!
Let's look at each one:

Centroid

Draw a line (called a "median") from a corner to the midpoint of the opposite side. Where all three lines intersect is the centroid, which is also the "center of mass":

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Circumcenter

Draw a line (called a "perpendicular bisector") at right angles to the midpoint of each side. Where all three lines intersect is the center of a triangle's "circumcircle", called the "circumcenter":

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Incenter

Draw a line (called the "angle bisector") from a corner so that it splits the angle in half Where all three lines intersect is the center of a triangle's "incircle", called the "incenter":

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Orthocenter

Draw a line (called the "altitude") at right angles to a side and going through the opposite corner. Where all three lines intersect is the "orthocenter":

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Note that sometimes the edges of the triangle have to be extended outside the triangle in order to draw the altitudes. Then the orthocenter is also outside the triangle. 

MathsIsFun