TOT 209 1989 Spring J3 convex ABCD , PQRS from paper and cardboard
Source:
March 7, 2021
geometrycombinatoricscombinatorial geometry
Problem Statement
The convex quadrilaterals and are made respectively from paper and cardboard. We say that they suit each other if the following two conditions are met :
( 1 ) It is possible to put the cardboard quadrilateral on the paper one so that the vertices of the first lie on the sides of the second, one vertex per side, and
(2) If, after this, we can fold the four non-covered triangles of the paper quadrilateral on to the cardboard one, covering it exactly.
( a) Prove that if the quadrilaterals suit each other, then the paper one has either a pair of opposite sides parallel or (a pair of) perpendicular diagonals.
(b) Prove that if is a parallelogram, then one can always make a cardboard quadrilateral to suit it.(N. Vasiliev)