# Activities

• • • ##### Subject Area

• Math: Geometry: Circles
• Math: Geometry: Triangles

• ##### Author 9-12
College

60 Minutes

• ##### Device
• TI-89 / TI-89 Titanium
• TI-92 Plus / Voyage™ 200
• ##### Software

Cabri Geometry™

## 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.

• 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