This page shows advanced examples of Cabri's use. Videos of basic classroom use of Cabri are at this link. Cabri's has great mathematical integrity due to the depth and quality of the underlying code. This is demonstrated by a wide range of examples impossible to create with such accuracy any other way. You can trust results from Cabri.
Morley’s triangle IJK is constructed using hyperbolic trisecting lines. The three angle values of the triangle are displayed.
They do not sum to 180 degrees, but this is normal since we are in Hyperbolic Geometry.
The three values are definitely different and the triangle is not an equilateral triangle, in contrast to the Euclidean case.
You can download the actual Cabri file for this illustration here and explore thedynamic aspects of the model.
Is an astroid made of arcs of ellipses?
Line segment AB, of constant length and with A on y-axis, models a ‘falling ladder’. The curve in red is the envelope of the supporting line, actually an astroid, apparently made of four parts, each being perhaps an arc of a circle or of an ellipse.
We can easily disqualify an arc of a circle, simply by looking. We can then test whether an ellipse fits our curve using Cabri’s conic tool selecting five points on the locus. Apparently the fit is excellent.
However, Cabri’s ‘Check Property’ feature proves that the fit is not exact: consider an additional point M on the locus and test its membership to the ellipse. Cabri reports that it is not.
The Witch of Agnesi
Fermat, and then Italian mathematician Agnesi, have studied the curve obtained from circle OA as shown on the left.
In this drawing smart lines have been used to focus the reader’s attention on the important aspects. Finally, the Witch of Agnesi is obtained as a locus of point P when M is moving on the circle.
Cabri’s equation feature can be applied to the locus and give an equation with integral parameters, here 16 and 64, matching the value 4 of the diameter of the circle. Varying the point A with integer steps, one can easily infer the general symbolic formula X[use formula at top of illustration] for the equation of the Witch of Agnesi. This is a unique feature offered by Cabri.
A formal description of the construction is given by “Option/Show Figure Description” - which is unique to Cabri. Highlight an element and the corresponding part of the formal description is highlighted. Moreover, only Cabri offers an automatically generated textual summary of each stage of the construction, which can be printed as a report.
• Cabri lets you replay a construction to see the sequence of actions taken. Use familiar transport controls like a tape recorder to step forwards or backwards throughout a construction. Clearly this helps teachers to see exactly which steps an individual student has taken, and to then go through discussing each step with the student.
• Cabri is fully interactive and dynamic geometry software available on handhelds, such as the TI-83+, TI-84 and TI Voyage 200. The CabriJunior application on these machines gives Cabri worldwide prevalence. The new TI-84 has superb screen resolution and is a very practical tool for exploring geometry.
• Cabri lets you redefine points or objects. This is clearly helpful in the development of models.
• Only Cabri lets you customise menus so that only tools relevant to the task appear, or to remove more complex possibilities to avoid confusion and guide thinking. CabriJava lets you publish dynamic constructions on web pages - see for example this kaleidoscope created by Kate Mackrell. See also Wilson Stother's Cabri and CabriJava pages.