Virginia Mathematics Teacher Spring 2017

From the numerical point of view it makes more sense to use an equivalent definition of a parallelogram, namely the congruence of opposite sides. This property is much easier to measure than parallelism of two objects (see Figure 3).

Fortunately, a few DGS currently include a feature closely related to mathematics reasoning: that of having implemented an Automated Theorem Proving (ATP) algorithm, yielding the ability to confirm or deny the mathematical (i.e. not numerical or probabilistic) truth of a geometric statement. In GeoGebra, some ATP features can also be introduced in the classrooms by using the Relation Tool. This tool in GeoGebra was originally designed to collect numerical equalities in a geometrical construction, but the recent versions can also be asked to re-investigate the problem symbolically (see Figure 5).

Figure 3 . Measuring the length of two opposite sides of a midpoint quadrilateral. Even if it is technically easier, the accuracy of such attempts is questionable. Even if it is performed on a graphing paper having a regular grid, the easy case with initial points (0,0), (2,0), (2,2) and (0,2) yield the midpoints (1,0), (2,1), (1,2) and (0,1) which result in an irrational length for each appearing side, namely the square root of 2. On accurate sampling when rounding the lengths of the opposite sides to 2 decimal places we will clearly get 1.41 for all four sides. By increasing the accuracy obviously the same results are expected. This is no longer true for some more complicated cases. By changing the rounding in DGS GeoGebra to 15 decimal places (in menu Options > Rounding > 15 Decimal Places), the last digits in the lengths of two opposite sides will differ (see Figure 4).

Figure 5 . Using GeoGebra's Relation Tool to numerically check properties (Kovács, 2016). When using the Relation Tool, the user needs to select two objects to compare. Despite the fact that the numerical computation is inaccurate, GeoGebra assumes that the opposite sides have the same length, and offers a further check to symbolically prove the congruence of those segments (see Figure 6).

Figure 4. A numerical error suggests falsifying a true statement.

Figure 6 . The Relation Tool provides a symbolical check to prove a property.

Virginia Mathematics Teacher vol. 43, no. 2

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