[alogo] 1. Parabola circumscribing trapezium

Here I study some properties connected to the parabola circumscribing a given trapezium RSS''R''. From the analysis given below results the construction of the unique parabola circumscribing the trapezium:
1) Locate Z which is harmonic conjugate of P' with respect to {R0,S0}.
2) Locate the middle T of P'Z and the symmetric S' of S0 w.r. to T.
3) Construct the parabola through {S,S''} tangent at these points respectively to lines {S'S, S'S''}.

[0_0] [0_1] [0_2] [0_3]
[1_0] [1_1] [1_2] [1_3]

[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 R0S0 joining the middles of the parallel chords. Similarly the tangents at {R,R''} intersect at R' on the same line. From elementary properties of trapezia we know that lines {RS, R''S''} intersect at a point Z also on line R0S0.

[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 R0S0 and is harmonic conjugate to Z with respect to {R',S'}.
First claim is a general property of trapezia. Second follows by projecting the cross ratio (R,S,P,Z)=-1 from V onto line R0S0. This indeed projects to (R',S',P',Z), which consequently is also -1.

[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 R0S0 = S'R', R0P' = R'Z and S0P' = S'Z.
This is a trivial consequence of the parabola property by which T is the middle of R0R' and also the middle of S0S', hence R0S0 = S'R'. The result follows from [4] by taking into account that P'R0/P'S0 = ZR0/ZS0 and ZS'/ZR' = P'S'/P'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.

[alogo] 2. Where is the second parabola

It is well known that given four points {A,B,C,D} there exist either two or none parabola passing through these points. This subject is studied in AllParabolasCircumscribed.html . Where is here the other parabola?
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 degnerate parabola passing through the four vertices of the trapezium.

See Also


Return to Gallery

Produced with EucliDraw©