Constraints and Constructions: Two Ways to Work

Constraints and Constructions: Two Ways to Work

Top 

Use Geometry Expressions to solve problems with a straightforward three-step process:

  1. Draw geometric objects such as points, lines, or circles. You need not draw them exactly — Geometry Expressions will make the necessary adjustments as you work.
  2. Constrain their relationships. As you specify constraints, the drawing adjusts to satisfy them. You can constrain the problem fully, or you can leave some elements unspecified.
  3. Request a measurement. Geometry Expressions adds any missing variables required for the calculation, then outputs the requested expression or value.

It’s not important to draw objects exactly at first; instead, the application corrects the drawing one step at a time as you define the problem by adding constraints.  This constraint-based approach enables a flexible, exploratory work style. It also readily lends itself to defining a problem symbolically, as it allows you to specify constraints — distances, angles, slopes, etc. — in terms of symbols.  Geometry Expressions provides a rich set of geometry objects and constraints, described in the User Interface Reference and embedded Help system.

You may be familiar with other interactive geometry applications, such as Geometer’s SketchPad® or Cabri Geometry™, which use a different approach:

  1. Draw independent geometric objects.
  2. Define dependent objects using constructions.

Geometry Expressions also enables this work style, providing a rich set of constructions, too (see the User Interface Reference and embedded Help system).

A construction defines a new geometry object in terms of existing objects. A construction-based approach requires you to distinguish between independent and dependent objects, and draw the independent objects accurately from the start. The drawing starts out correct and remains correct at each step. To enable this, the approach typically requires some foreknowledge of the problem geometry.

To compare these two approaches, we’ll use both to create a drawing that shows the incircle of a triangle — the circle inside the triangle that’s tangential to all three sides.