Different approaches of interplay between experimentation and theoretical consideration in dynamic geometry exploration: An example from exploring Simon line. Yip-Cheung C HAN Faculty of Education The University of Hong Kong Hong Kong.
Different approaches of interplay between experimentation and theoretical consideration in dynamic geometry exploration: An example from exploring Simon line
Faculty of Education
The University of Hong Kong
The author acknowledges the financial supports of HKU CRCG Conference Grants for Research Students & Sik Sik Yuen Education Research Fund.
ABC is a triangle on a plane. P is an arbitrary point on that plane. Let X, Y, Z be the feet of the perpendicular lines drawn from P to the sides AB, BC and AC respectively. (Sides AB, BC and AC can be extended if necessary.)
Consider triangle ABC. Let P be a point and X, Y, Z be the feet of perpendicular lines drawn from P to the (extended) sides AB, BC and AC respectively. The locus of P such that X, Y, Z are collinear is the circumscribed circle of triangle ABC.
Posamentier, A. S. (2002). Advanced Euclidean Geometry: Excursions for Secondary Teachers and Students. Emeryville, California: Key College Publishing.
A line joining X & Z was constructed. This served as a “guide” for locating suitable positions of P.
P was randomly dragged until Y lied on that line (drag-to-fit strategy)
Trying to “mark” the suitable positions by “trace” (drag-to-fit strategy)
The “trace” method seems not satisfactory. A new method of marking the found positions was invented.
Based on her intuition, she guessed that the locus is a circle. But what kind of circle is it?
She constructed the circumscribed circle by locating the intersection of perpendicular bisectors (circumcenter) and verified that all marked points lied on this circle.
P was merged onto the circumscribed circle. Animation was used to show that the required locus is the circumscribed circle, as expected.
Another line joining X, Y was constructed to ensure X, Y, Z were collinear.
led to inventing a new method
P was accidentally positioned at vertex C
led to making the conjecture (the locus is the circumscribed circle)
Constructed 2 lines and checked whether they were overlapped.
A line joining X & Y was constructed. This served as a consideration in Tracy’s exploration“guide” for locating suitable positions of P. Then, P was randomly dragged until Z lied on that line. This method was repeated to locate other positions of P.
Guess 1: XP is parallel to BC screen.
Guess 2: XPYB is concyclic screen.
Trying to find the radius. the sides of the triangle [in this case, side AB]. But, by
Finally, he concluded that the locus should be the circumscribed circle of triangle ABC.
This led him to switch the path of exploration (coordinate geometry)
As different people use interactive software differently, flexibility for catering individual differences should be one of the major concerns in future development of interactive software. (e.g. “put a nail on the position”)