1. introduction
1.1. geometrical constructions comprise a fascinating and surprising world which has been the focus of attention of mathematicians since ancient times. As time went on, occasionally new methods were found to carry out additional constructions possible. among properties.
1.1.1. methodical note
1.1.1.1. the constructions are made possible by employing the special properties of the loci. only when we met a dead end in proceeding with our constructions we turned to the textbooks and found in them unknow properties using which we were successful in dealing with the challenge pf construction.
2. using a dynamic geometry environment (DGE)
2.1. students/learners can experiment through different dragging modalities on geometrical objects that they construct, and consequently infe properties, generalities, and conjectures about the geometrical artifacts.
2.1.1. summary
2.1.1.1. different methods were proposed to carry out geometrical constructions in the context of there loci that appear in the program of studies of high school education.
2.1.2. acknowledgements
2.1.2.1. the authors of the article, thank very much to Mr. Idan Tak for the great assistance in preparing the geogebra applets
3. link 1, A: straight line that connects the middles of parallel chords in a parabola https://www.geogebra.org/m/csdNrzft
3.1. Applet 1, that can be reached by link 1, ilustrates the property that the straight line that passes through the midpoint of a pair of parallel chords is parallel to the axis of the parabola. by dragging the point C one can change the distance between the parallel chords.
3.1.1. References
3.1.1.1. [1] Henkin, L. and Leonard, W. A., 1978: A Euclidean Construction? Mathematics Magazine, 51(5), 294–298. [2] Downs, J. W., 1993: Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas. Courier Co. [3] Riddle, D. R., 1996: Analytic Geometry, 6th ed., PWS-Kent Publishing Co. [4] Stupel, M. and Ben-Chaim, D., 2013: A fascinating application of Steiner’s Theorem for Trapezoids-Geometric constructions using straightedge alone. Australian Senior Mathematics Journal (ASMJ), 27(2), 6–24. [5] Stupel, M., Oxman, V. and Sigler, A: More on Geometrical Constructions of a Tangent to a Circle with a Straightedge only. The Electronic Journal of Mathematics and Technology. 8(1), 17–30. [6] Alperin, R. C., 2016: Axioms for Origami and Compass Constructions. Journal for Geometry and Graphics. 20(1), 13–22.
3.1.1.2. [7] Yiu, P., 2014: Three Constructions of Archimedean Circles in an Arbelos. Forum Geometricorum. 14, 255–260. [8] Yiu, P., 2012: Conic Construction of a Triangle from its Incenter, Nine-point Center, and a Vertex. Journal for Geometry and Graphics. 16(2), 171–183. [9] Dergiades, N., 2015: Construction of Ajima Circles via Centers of Similitude. Forum Geometricorum. 15, 203–209. [10]. De Villiers, M., 1998: An alternative approach to proof in dynamic geometry. Designing Learning Environments for Developing Understanding of Geometry and Space, R. Lehrer and D. Chazan, Eds., Lawrence Erlbaum Associates, Hillsdale, NJ, 369–394. [11] Leung, A. and Lopez-Real, F. J., 2002: Theorem justification and acquisition in dynamic geometry: A case of proof by contradiction. International Journal of Computers for Mathematical Learning, 7(2), 145–165. [12] Liekin, R., 2009: Multiple proof tasks: Teacher practice and teacher education, ICME Study 19, Vol. 2, 31–36.