Corollary: The diagonals of all quadrilaterals HIJK intersect at E.

In fact, N, M can be taken as the intersection points of opposite sides of q. Then N is on the polar of E, hence the polar p(N) of N contains E. Consider the intersection points O, P of this polar with sides HK, IJ respectively. Assume also L to be the intersection point of HI, DC. Then a) these sides intersect at a point L lying on p(N). b) L is on line MN. a) follows from the standard theorem on cyclic quadrilaterals (see CyclicProjective.html ). b) follows from the fact that the quadrupple of lines at N (NL,NH,NE,NI) is harmonic. But (NM,NH,NE,NI) is also harmonic, hence L is contained in line MN.

The corollary follows also from the fact that for cyclic quadrilaterals q=ABCD the diagonals intersect at the pole E of MN (see reference above).

Produced with EucliDraw© |