a) Three of six segments (four sides and two diagonals of an isosceles trapezoid) are painted red, and three others are painted green.
Prove that one can construct a triangle using the segments of the same color as its sides.
b) Is the previous statement true if these six segments are four sides and two diagonals of an arbitrary trapezoid?
b) The statement may not hold.
a) See the solution of Problem D.4.
b) Consider, for example, a trapezoid such that its diagonal is perpendicular to the bases and (see the Fig.). Let . Let be painted green and be painted red. For these red segments the following inequality holds
(see the Fig.), so one cannot construct the triangle with these red segments as its sides.
We select the lengths of the bases , so that one cannot construct the triangle with these green segments as its sides. It suffices to choose and such that the following inequality