Students understand that the orthocenter is the point of concurrence of the three altitudes of a triangle. They construct circumcircles and Feuerbach circles and determine the relationship between them.

Before the Activity

See the attached PDF file for detailed instructions for this activity

Print page 15 from the attached PDF file for your class

During the Activity

Distribute the page to the class.

Follow the Activity procedures:

Draw a triangle, construct the altitudes and locate the orthocenter of the triangle

Alter the triangle and observe that for an acute triangle, the orthocenter lies in the interior of the triangle, for obtuse triangle, it lies outside the triangle and for right triangle, it lies at the vertex of the right angle

Hide the altitudes and the triangle, so that only the vertices and the orthocenter remain on screen

Notice that the triangle formed by any combination of three of these points has the fourth point as its orthocenter

Construct all possible triangles formed by these points, and draw the 9-point circle

Construct the circumcircles of all the triangles

Note that the areas of the circumcircles are equal, and the area of the 9-point circle is one-fourth the area of any one of the circumcircles

Hide all the circles, and connect the circumcenters of the triangles

Notice that a figure congruent to the original figure is formed

Understand that any triangle in the new figure has the same nine-point circle as any triangle in the original figure

Observe that the center of the nine-point circle represents the center of rotation that transforms the original figure into the new figure

After the Activity

Review student results:

As a class, discuss questions that appeared to be more challenging