Where is the center of a triangle?
Surprisingly, there can be five different answers!
The first can be found by drawing on the three perpendicular bisectors, at right angles from the midpoint of each side.
The lines cross at the circumcenter of the triangle.
To find another center, draw a triangle's medians.
These are lines from its corners to the midpoints on opposite sides.
The point where the three medians intersect is called the centroid.
For another center, draw in the three angle bisectors that divide the angles equally in two.
This center point is called the incenter.
For a fourth center, draw a triangle's three altitudes.
These are straight lines running from the vertices, or corners, which meet opposite sides at right angles.
Their intersection is called the orthocenter.
A fifth center for a triangle is found by drawing a circle.
It should touch the midpoints of each side.
The foot of each altitude.
And the point on each altitude, midway between the corner and the orthocenter.
The center of the circle these nine points create is called the nine-point center.
Within any given triangle, the orthocenter, the nine-point center, the centroid, and the circumcenter lie on one straight line.
This is called Euler's line.
And in an isosceles triangle, the incenter fits as well; meaning Euler's line unites all five possible centers of the triangle.