Education Technology

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

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