\input zb-basic \input zb-matheduc \iteman{ZMATH 2016f.01013} \itemau{Fraivert, David} \itemti{Discovering new geometric properties by spiral inductive-deductive investigation.} \itemso{Far East J. Math. Educ. 16, No. 2, 185-202 (2016).} \itemab 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. \itemrv{~} \itemcc{G40} \itemut{induction; deduction; trapezoid properties; quadrilateral and circle; applications of the generalized Pascal theorem; pole and its polar with respect to a circle; harmonic quadruple of four points on a straight line; GeoGebra software} \itemli{http://www.pphmj.com/abstract/9770.htm} \end