## Centers of a Triangle

### Sue Meredith

The Geometer's Sketchpad gives the user the facility to analyze many different aspects of triangles that become obvious with this tool. Doing the same constructions with compass and straightedge would be time consuming and without the accuracy would not demonstrate the concepts as clearly. In my class with 7th graders, we have constructed the incenter with straightedge and compass with some degree of success; however the advantages of the GSP construction are obvious.

To begin the study of the centers of a triangle, we start with the centroid by constructing the medians, with connecting the vertices to the opposite midpoints.

On GSP we can move any vertex to see the movement of the centroid with the changes in the triangle.

The centroid does move but not dramatically as the triangle is changed from acute to right and to obtuse.

The second triangle construction is the orthocenter. The orthocenter is the point of intersection of the altitudes of the triangle.

When the triangle is changed to right and obtuse the orthocenter moves radically.

When the triangle is right, H is on the vertex of the right angle, and with an obtuse triangle, the orthocenter is outside the triangle.

The circumcenter is the third construction. The circumcenter is equidistant from the three vertices and therefore is the center of a circle that circumscribes the triangle. To find that point, construct the perpendicular bisector of each side. The circumcenter is the center of the circmcircle as shown ineach of the following.

Once again, we change the triangle to see the effect on the circumcenter.

On a right triangle the circumcenter is on the hypotenuse, and on an obtuse triangle it is outside the triangle.

The fourth center is the incenter, constructed by bisecting the angles to find points equidistant from the sides. The incenter is the center of the inscribed circle of the triangle.

Changing the shape of the triangle produces the following changes in the incenter.

When G,H and C are all constructed in one triangle, they are collinear. This line is Euler's line, shown here in blue. Changing the triangles demonstrates some of the qualities of Euler's line; for example, in the right triangle it is the median.

The following show the consistency of Euler's line.

When the Medial triangle is constructed by connecting the midpoints of the triangle and the G, H and C are constructed for the new triangle the line they coincide with the ones for the original.

The following triangle has the Medial triangle in green, formed by connecting the midpoints of the original triangle. The incircle in red, and Euler's line in blue.

The Nine-point circle can be constructed by finding midpoints of the sides, points at the feet of the altitudes and midpoints of the segments from orthocenter to the respective vertices. In order to find the center of the nine-point circle, use the segments for diameters, the intersection gives the center and a radius for the circle.

There are many more investigations of triangles in Assignment 5, where scripts are written to repeat these constructions automatically.

Proof:

Given: any triangle, with 3 perpendicular bisectors.
To Prove: The 3 perpendicular bisectors are concurrent.