1) Locate Z which is harmonic conjugate of P' with respect to {R

2) Locate the middle T of P'Z and the symmetric S' of S

3) Construct the parabola through {S,S''} tangent at these points respectively to lines {S'S, S'S''}.

[1] Given the parabola (c) consider two parallel chords {S,S'',R,R''}. From elementary properties of the parabola we know that the tangents {SS', S''S'} at {S,S''} intersect at a point S' on the line R

[2] If {V,V'} are the intersections of lines (RR',SS') and (R'R'',S'S''}. Line VV' is the polar of Z.

In fact, Z is on the polar of V hence, by the duality of poles-polars, V is also on the polar of Z. Analogously V' is on the polar of Z, hence VV' is the polar of Z.

[3] Let P' be the intersection point of the diagonals of the trapezium RSS''R''. Then P' is on R

First claim is a general property of trapezia. Second follows by projecting the cross ratio (R,S,P,Z)=-1 from V onto line R

[4] Let points {U, U'} be the intersections of line-pairs (RR',S'S'') and (SS',R'R'') respectively. Point Z is the intersection point of the diagonals of quadrilateral S'UR'U'. Diagonal UU' is parallel to VV'.

First claim follows from the general fact on harmonic divisions through diagonals of quadrilaterals and the previous paragraph. The same reason implies also the second claim. In fact VV' and UU' intersect at a point W such that (V,V',P',W)=(U,U',Z,W) = -1. Since P' is the midde of VV' W is at infinity, hence VV', UU' are parallel.

[5] Define projectivity F by requiring that {F(S) = S', F(S') = S'', F(R) = R', F(R') = R''}. Then Z is a fixed point of F, line PP' is invariant under F and V maps to V'.

In fact, by elementary properties of the parabolas, Z is intersection point of lines {RS, R'S'} which map to {R'S', R''S''} hence define the same intersection point.

To prove the second claim notice that since line RS maps to line R'S' and projectivities respect cross-ratios F(P) must satisfy (R,S,P,Z) = (R',S',F(P),Z) = -1, which by [4] identifies F(P) with P'. Analogously one shows that F(P') = P'' and consequently the invariance of line PP' by F.

Last claim is a consequence of the definition.

[6] Let T the point on the parabola with tangent there parallel to the parallels {SS'', RR''}. Then R

This is a trivial consequence of the parabola property by which T is the middle of R

[7] The projectivity F is not an affinity (does not leave invariant the line at infinity).

In fact, the point I which is the image under F of the point at infinity represented by the parallel lines VV' and UU' is a point other than infinity of the invariant line VV'. This point can be even determined from the invariance of cross-ratio. In fact, the ratio PP'/PV coincides with the cross ratio (P',V,P,inf), where (inf) represents the point at infinity of line VV'. By the invariance of cross-ratio under F we have (P'',V',P',I) = PP'/PV, hence (P'P''/P'V'):(IP''/IV') = PP'/PV. From P'P''=PP' follows IP''/IV' = PV/P'V' = - P''V'/P'V'. Thus I is a point inside V'P'' that can be located from the ratio of the distances from points {V',P''}.

[8] Analogously to F one can define another projectivity F' by prescribing the values {F'(S) = S', F'(S') = S'', F'(R'') = R', F(R') = R}. In this case one shows analogously that P' is fixed point of F' and line UU' is invariant by F'. Also in this case F' is not an affinity.

The answer is: look at the two parallel lines carying the parallel sides of the trapezium. In fact, the union of these parallel lines represents a

AllParabolasCircumscribed.html

Produced with EucliDraw© |