Circumcenter, Incenter, Orthocenter vs Centroid
Circumcenter: circumcenter is the point of intersection of three perpendicular bisectors of a triangle. Circumcenter is the center of the circumcircle, which is a circle passing through all three vertices of a triangle.
To draw the circumcenter create any two perpendicular bisectors to the sides of the triangle. The point of intersection gives the circumcenter. A bisector can be created using the compass and the straight edge of the ruler. Set the compass to a radius, which is more than half the length of the line segment. Then make two arcs on either side of the segment with an end as the center of the arc. Repeat the process with the other end of the segment. The four arcs create two points of intersection on either side of the segment. Draw a line joining these two points with the aid of the ruler, and that will give the perpendicular bisector of the segment.
To create the circumcircle, draw a circle with the circumcenter as the center and the length between circumcenter and a vertex as the radius of the circle.
Incenter: Incenter is the point of intersection of the three angle bisectors. Incenter is the center of the circle with the circumference intersecting all three sides of the triangle.
To draw the incenter of a triangle, create any two internal angle bisectors of the triangle. The point of intersection of the two angle bisectors gives the incenter. To draw the angle bisector, make two arcs on each of the arms with the same radius. This provides two points (one on each arm) on the arms of the angle. Then taking each point on the arms as the centers, draw two more arcs. The point constructed by the intersection of these two arcs gives a third point. A line joining the vertex of the angle and the third point gives the angle bisector.
To create the incircle, construct a line segment perpendicular to any side, which is passing through the incenter. Taking the length between the base of the perpendicular and the incenter as the radius, draw a complete circle.
Orthocenter: Orthocenter is the point of intersection of the three heights (altitudes) of the triangle.
To create the orthocenter, draw any two altitudes of a triangle. A line segment perpendicular to a side passing through the opposing vertex is called a height. To draw a perpendicular line passing through a point, first mark two arcs on the line with the point as the center. Then, create another two arcs with each of the intersection points as the center. Draw a line segment joining the first point and the finally constructed point, and that gives the line perpendicular to the line segment and passing through the first point. The point of intersection of the two heights gives the orthocenter.
Centroid: Centroid is the point of intersection of the three medians of a triangle. Centroid divides each median in 1:2 ratio, and the center of mass of a uniform, triangular lamina lies at this point.
To determine the centroid, create any two medians of the triangle. For creating a median, mark the midpoint of a side. Then construct a line segment joining the midpoint and the opposing vertex of the triangle. The point of intersection of the medians gives the centroid of a triangle.
What are the differences among Circumcenter, Incenter, Orthocenter and Centroid?
• Circumcenter is created using the perpendicular bisectors of the triangle.
• Incenters is created using the angles bisectors of the triangles.
• Orthocenter is created using the heights(altitudes) of the triangle.
• Centroid is created using the medians of the triangle.
• Both the circumcenter and the incenter have associated circles with specific geometric properties.
• Centroid is the geometric center of the triangle, and its is the center of mass of a uniform triangular laminar.
• For a non equilateral triangle, the circumcenter, orthocenter, and the centroid lies on a straight line, and the line is known as the Euler line.