In this paper, we have considered the lines that bisect the area and/or perimeter of the triangle. It has been shown that in any triangle there are infinitely many bisectors of area and infinitely many bisectors of the perimeter. These pairs of lines form envelopes called deltoids because of their shape. Deltoids are described in the third chapter, where some interesting properties are also shown. The lines that are both perimeter bisectors and area bisectors are called AP-bisectors, and they are of particular interest. There is a finite number of AP-bisectors in each triangle, and this number depends on the relations between the sidelengths of the triangle. Specifically, the number of AP-bisectors depends on the sign of the discriminant of a quadratic equation whose real solutions are the length intercepts formed by the bisector on the sides of the triangle with respect to one of the vertices. A discussion of the solutions shows that for each triangle there is one, two or three AP-bisectors depending on the sign of the discriminant. Possible cases of triangles and their number of AP-bisectors are shown by examples. After proving the results about bisectors, a construction of such lines is shown. Furthermore, an additional analysis including results from number theory is carried out, in order to produce sequences of triangles whose sidelengths, as well as the aforementioned lengths of intercepts formed by AP-bisectors are integers.