True or false? In every triangle, the centroid, incenter, and circumcenter are collinear.
false
False. In every triangle, the centroid, incenter, and circumcenter are not collinear.
To understand why, we need to first define what each of these points represent in a triangle:
1. Centroid: The centroid of a triangle is the point of intersection of the medians, which are the lines joining each vertex of the triangle to the midpoint of the opposite side.
2. Incenter: The incenter of a triangle is the point of intersection of the angle bisectors, which are the lines that divide each of the triangle's angles into two equal parts.
3. Circumcenter: The circumcenter of a triangle is the point of intersection of the perpendicular bisectors, which are the lines that bisect each side of the triangle at a 90-degree angle.
Now, to determine whether these points are collinear, we can use a counterexample.
Consider an equilateral triangle, where all three sides are equal and all angles are 60 degrees. In this case, the centroid, incenter, and circumcenter do not coincide and are not collinear.
Thus, the statement "In every triangle, the centroid, incenter, and circumcenter are collinear" is false.