Interactive or Dynamic Geometry System (DGS) is a tool that help to teach and learn geometry using a computer-based interactive environment. Traditionally, the interaction with DGS is based on keyboard and mouse events where the functionalities are accessed using a menu of icons. Nevertheless, recent findings suggest that such a traditional model of interaction has a steep learning curve and is inadequate to develop DGS for devices with multi-touch screens. Thus, we propose a new interaction model for DGS based on a gesture dictionary which enables the construction and manipulation of geometric objects without the need of accessing a menu of icons. The dictionary is divided into three types: (i) kernel gestures: which are the basis for defining gestures; (ii) navigation gestures: related to the manipulation and editing of geometric objects; and (iii) basic gestures of construction: to construct geometric objects. To validate our interaction model, a DGS for mobile device, GeoTouch, has been developed using the proposed dictionary. Usability tests were performed comparing GeoTouch with three other DGS for mobile devices. The results indicate that the GeoTouch interface has fewer usability problems than the other three DGS software and it is easier to learn and interact with.