Dear Hauke, I think the answer is yes.

If we consider an arbitrary convex quadrilateral ABCD, we can construct

a point P inwardly to it, such that the triangles PAB, PDC to be

similar, with for example <PAB = <PCD and <PBA = <PDC, as well known.

( The point P is the intersection point of two Appolonian circles with

respect to the diagonals AC, BD of ABCD and ratio r = AB / CD ).

Let now S be, the intersection point of the sidelines AD, BC and we

define three points P', C', D', on the rays SP, SC, SD respectively,

such that P'C' // PC and P'D' // PD.

The triangles PAB, P'D'C', are perspective and similar as the problem

states.

Best regards, Kostas Vittas.

--- In Hyacinthos@yahoogroups.com, "Hauke Reddmann" <fc3a501@...> wrote:

>

> Let ABC and A'B'C' be perspective trangles.

> Can ABC and A'B'C' be similar *without*

> AB||A'B' etc.?

>

> Hauke

>