1. Preserving relative area of a quadrilateral in a triangle
a pencil of three straight lines is drawn from the vertex A, and a pencil of two lines issues from the vertex C. After the students were instructed and had basic hands-on experience with regards to the possibilities of the dynamic geometric software, a dynamic applet was prepared for them in which the vertices of the triangles can be dragged, thereby varying the lengths of its sides and changing its area. In addition, the applet includes data bars using which the values of m and n can be changed – these values represent the numbers of the straight lines in the pencils that issue from the vertices A and C. At each stage, the area of the quadrilateral by the vertex B (marked by poly1), of the triangle ABC (marked by poly2) and the ratio between the areas (marked by p) appear on the screen. At the first stage the students constructed examples of particular cases, such as m=1, n=1, which corresponds to the construction of two medians in the triangle. For this case they obtained that the area ratio is always 1/3, and does not depend on the side lengths of the triangle.