Activity 9
Special Properties of Polygons

There are some special properties of triangles and quadrilaterals that can be investigated. These investigations require the use of a compass, Mira, and straightedge.

Special Points in Triangles

Centroid The median of a triangle is a segment connecting a vertex to the midpoint of the opposite side. The point of intersection of the three medians of a triangle is called the centroid of the triangle. The centroid is located at the center of gravity of the triangle. If the triangle were made of a constant thickness material of a constant density, it could be balanced at this point. Find the centroid of the triangle below by drawing the three medians.

Incenter The three angle bisectors of a triangle intersect at a point that is called the incenter of the triangle. It is called the incenter because it is the center of the largest circle that will fit inside the triangle. This circle will be tangent to all the sides of the triangle. Find the incenter of the triangle below by drawing all three angle bisectors. Using the incenter as the center, draw the circle that is tangent to all of the sides of the triangle.

Circumcenter The perpendicular bisector of a side of a triangle may or may not pass through the opposite vertex. The intersection of all three perpendicular bisectors of the sides of a triangle is called the circumcenter of the triangle. This point is called the circumcenter because it is the center of a circle that passes through all of the vertices of the triangle. Using this technique, a circle can be drawn so that it passes through any three non-collinear points on a plane; simply connect the three points to form a triangle; then find the circumcenter and draw the circle. Find the circumcenter of the triangle below by drawing all three perpendicular bisectors. Using the circumcenter as center, draw the circle that passes through all of the vertices of the triangle.

Find the circumcenter of the obtuse triangle below by drawing all three perpendicular bisectors. Note that the circumcenter lies outside of the triangle. Using the circumcenter as center, draw the circle that passes through all of the vertices of the triangle.

Orthocenter An altitude of a triangle is a segment that is drawn so that it passes through a vertex and is perpendicular to the opposite side. The intersection of the three altitudes of a triangle is called the orthocenter. Find the orthocenter of the triangle below by drawing the three altitudes.

The obtuse triangle below also has an orthocenter, but it lies outside of the triangle. Find the orthocenter by drawing the three altitudes. One of the altitudes will pass through the triangle. The other two altitudes will lie completely outside of the triangle. To draw an altitude that lies outside the triangle, it is necessary to extend two of the sides of the triangle.

Eüler Line The centroid, circumcenter, and orthocenter of a triangle are all collinear. The line that includes these three points is called the Eüler Line. Draw the Eüler Line for the triangle below by finding the centroid, circumcenter, and orthocenter of the triangle. What relationship is there between the distances from the centroid to the orthocenter and the orthocenter to the circumcenter?

Midpoints of Sides

Find the midpoints of the sides of the triangles below and connect them to form new triangles. Notice that these triangles separate the original triangles into four triangles. What do you notice about these four triangles?

How can you support this statement?

Find the midpoints of the sides of the quadrilaterals shown below and connect these midpoints to form another quadrilateral. What do you notice about these quadrilaterals that are formed?

How can you support this statement?

Find the midpoints of the sides of the square shown below. Connect these midpoints to form a second square. How is the area of this square related to the area of the original square? Repeat this process five more times, forming a third, fourth, fifth, sixth, and seventh square. If the area of the first square is 1, what are the areas of each of the other squares? Place your answers in the table below. What is the area of the n^th square formed?

Square	1	2	3	4	5	6	7	n
Area	1	_____	_____	_____	_____	_____	_____	_____

Find the midpoints of the sides of the triangle shown below. Connect these midpoints to form a second triangle. How is the area of this triangle related to the area of the original triangle? Repeat this process five more times, forming a third, fourth, fifth, sixth, and seventh triangle. If the area of the first triangle is 1, what are the areas of each of the other triangles? Place your answers in the table below. What is the area of the n^th triangle formed?

Triangle	1	2	3	4	5	6	7	n
Area	1	_____	_____	_____	_____	_____	_____	_____