Education Technology

A special attribute of triangles with their vertices on an equilateral hyperbola (C4)

Published on 06/09/2008

Activity Overview

This activity illustrates the multiple representations of the attributes of triangles with their vertices on an equilateral hyperbola analytical, graphical and dynamic; addressing various conceptual levels of the students using CAS.

Before the Activity

  • Install the Cabri Geometry™ software on the computer
  • See the attached PDF file for detailed instructions for this activity
  • Print pages 1 - 4 from the attached PDF file for the class
  • During the Activity

    Distribute the pages to the class.

    Follow the Activity procedures:

  • Take any three points A, B, C on an equilateral hyperbola and find the orthocenter O of the triangle ABC
  • Note that the orthocenter is always a point on the hyperbola
  • Take any randomly chosen point on a random equilateral hyperbola and test this feature once more
  • Find a formula for the coordinates of the orthocenter of a triangle
  • Recognize a pattern in the results
  • Give a graphic representation of the triangle, the hyperbola, and the orthocenter
  • Find the circumcircle of the given triangle
  • Add the graph of this circle and point O which lies symmetric to orthocenter O with respect to the origin
  • Describe your results
  • Prove your conjecture
  • After the Activity

    Review student results.

  • As a class, discuss questions that appeared to be more challenging
  • Re-teach concepts as necessary