3. If the base of a triangle be divided into any number of equal parts, right lines drawn from the vertex to the points of division will divide the whole triangle into as many equal parts.
4. Right lines from any point in the diagonal of a parallelogram to the angular points through which the diagonal does not pass, and the diagonal, divide the parallelogram into four triangles which are equal, two by two.
5. If one diagonal of a quadrilateral bisects the other, it also bisects the quadrilateral, and conversely.
6. If two △s ABC, ABD be on the same base AB, and between the same parallels, and if a parallel to AB meet the sides AC, BC in the point E, F; and the sides AD, BD in the point G, H; then EF = GH.
7. If instead of triangles on the same base we have triangles on equal bases and between the same parallels, the intercepts made by the sides of the triangles on any parallel to the bases are equal.
8. If the middle points of any two sides of a triangle be joined, the triangle so formed with the two half sides is one-fourth of the whole.
9. The triangle whose vertices are the middle points of two sides, and any point in the base of another triangle, is one-fourth of that triangle.
10. Bisect a given triangle by a right line drawn from a given point in one of the sides.
11. Trisect a given triangle by three right lines drawn from a given point within it.
12. Prove that any right line through the intersection of the diagonals of a parallelogram bisects the parallelogram.