Education Technology

The Orthocenter of a Triangle

Published on 06/09/2008

Activity Overview

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
  • Re-teach concepts as necessary