Problem in Geometry, with Four Solutions, One of Them with Dynamic Research

Problem solving and proofs have always played a major role in mathematics. They are, in fact, the heart and soul of the discipline. The using of a number of different proof techniques for one specific problem can display the beauty, and elegance of mathematics. In this paper, we present one specific, interesting geometry problem, and present four different proofs for it, using methods, including using GeoGebra, that is dynamic geometry software (DGS) application, to make the initial conjecture. Encouraging students to derive results in such various ways will enhance their appreciation of mathematics, give them incentive to derive even more elegant solutions on their own, and provide them the opportunity to come up with some original insights on how proofs work. As a bonus, the problem we used, also let them know Ptolemy's theorem and the fourth theorem of the overlap of triangles, which are not included in the curriculum of some countries. The implication of our findings is that mathematics educators should be encouraged to introduce many such authentic multiple-proof problems into the teaching program. In addition, the effect of such exercises on students' mathematical understanding should be studied.