Parabolas Related to a Triangle

Extract

Let ABC be a plane triangle with D, E, F the midpoints of the sides BC, CA, AB respectively. Let L, L’ be two real intersecting straight lines in the plane. Then it can easily be seen, as in § 1 below, that if parallelograms are described on the three sides as diagonals, each with its sides parallel to L, L’, then the other diagonals of these parallelograms are concurrent at a finite point of the plane, which may be denoted by P(L, L’). An attempt to determine those points of the plane which are such points of concurrence has led to the following results.