Orthocenter:
1. Extend each side until it meets altitude from opposite vertex.
2. Find the intersection point of those three altitudes.
Centroid:
1. Divide each median into thirds.
2. Find the intersection point of those three medians.
Circumcenter:
1. Draw perpendicular bisectors of each side.
2. Find the intersection point of those three bisectors.
Incenter:
1. Draw angle bisectors of each angle.
2. Find the intersection point of those three bisectors.
The image you sent is a prompt to sketch the orthocenter, centroid, circumcenter, and incenter of a triangle. Here are the steps on how to do each:
Orthocenter:
1. Draw the triangle ABC.
2. Extend each side of the triangle beyond the vertex until it meets the perpendicular line drawn from the opposite vertex. For example, to find the orthocenter of vertex A, extend lines BC and AC beyond C and A, respectively, until they meet the perpendicular line drawn from vertex A.
3. The point where the three perpendicular lines intersect is the orthocenter of the triangle.
Centroid:
1. Draw the triangle ABC.
2. Divide each median of the triangle (a line segment from a vertex to the midpoint of the opposite side) into three segments. The centroid is the point where the three medians intersect.
3. The centroid is two-thirds of the way from each vertex to the midpoint of the opposite side.
Circumcenter:
1. Draw the triangle ABC.
2. Draw the perpendicular bisectors of each side of the triangle. The perpendicular bisector of a side is a line that passes through the midpoint of the side and is perpendicular to it.
3. The point where the three perpendicular bisectors intersect is the circumcenter of the triangle.
4. The circumcenter is the center of the circle that passes through all three vertices of the triangle.
Incenter:
1. Draw the triangle ABC.
2. Draw the angle bisectors of each angle of the triangle. An angle bisector is a line segment that divides an angle into two congruent angles.
3. The point where the three angle bisectors intersect is the incenter of the triangle.
4. The incenter is the center of the circle that is tangent to all three sides of the triangle.