Discovering new geometric properties by spiral inductive-deductive investigation. (English)

Far East J. Math. Educ. 16, No. 2, 185-202 (2016).

Summary: Induction and deduction are important tools in scientific investigation. In the present paper, we suggest a geometric investigation that describes a spiral bidirectional process of induction and deduction. The inductive process starts from a geometric situation that contains an arbitrary trapezoid and arrives at a hypothesis concerning the existence of a new property in the trapezoid.{ }The deductive process of proving this property leads to a new geometric situation that contains a trapezoid and the circle of Apollonius associated with that trapezoid. In this case, the inductive process leads to new hypotheses. The attempt at proving them leads to a more general geometric situation that contains a convex quadrilateral and a circle that passes through the points of intersection of the diagonals and the extensions of two opposite sides of that quadrilateral.{ }In this geometric situation, the mentioned hypotheses are proven, but the investigation does not end at this stage, since after expanding the last geometric situation into a broader one that also includes tangents to the circle, the investigation executes another inductive-deductive cycle of finding and proving properties of the tangents that have to do with the previous hypotheses.{ }In the performed investigation, we have discovered and proved a total of four new properties in Euclidean geometry concerning: (i) An arbitrary trapezoid; (ii) A quadrilateral and a circle that have special relationships between them.