Final answer:
In an isosceles triangle, the centroid, incenter, orthocenter, and circumcenter are collinear along the Euler line. The centroid is the intersection of the medians, the incenter is the intersection of the angle bisectors, the orthocenter is the intersection of the altitudes, and the circumcenter is the intersection of the perpendicular bisectors.
Step-by-step explanation:
An isosceles triangle has two sides that are equal in length. In such a triangle, the centroid, incenter, orthocenter, and circumcenter are collinear along a single line called the Euler line.
The centroid, denoted by G, is the point of intersection of the medians of the triangle. The medians are the line segments that connect each vertex to the midpoint of the opposite side.
The incenter, denoted by I, is the point of intersection of the angle bisectors of the triangle. The angle bisectors are the lines that divide each angle into two equal parts.
The orthocenter, denoted by H, is the point of intersection of the altitudes of the triangle. The altitudes are the lines that are perpendicular to a side of the triangle and pass through the opposite vertex.
The circumcenter, denoted by O, is the point of intersection of the perpendicular bisectors of the sides of the triangle. The perpendicular bisectors are the lines that bisect each side and are perpendicular to the side.