230 4 Geometry and Trigonometry
A
E
B C
D
Figure 33If all three segmentsAB,AC, andADhad rational lengths, this relation would imply
that
√
3 is rational, which is not true. Hence at least one of these lengths is irrational.646.Three lines passing through an interior point of a triangle and parallel to its sides
determine three parallelograms and three triangles. IfSis the area of the initial
triangle andS 1 ,S 2 , andS 3 are the areas of the newly formed triangles, prove that
S 1 +S 2 +S 3 ≥^13 S.
647.Someone has drawn two squares of side 0.9 inside a disk of radius 1. Prove that
the squares overlap.
648.A surface is generated by a segment whose midpoint rotates along the unit circle in
thexy-plane such that for each 0≤α< 2 π, at the point of coordinates(cosα,sinα)
on the circle the segment is in the same plane with thez-axis and makes with it an
angle ofα 2. This surface, called aMöbius band, is depicted in Figure 34. What is
the maximal length the segment can have so that the surface does not cross itself?
Figure 34649.LetABCDbe a convex quadrilateral and letObe the intersection of its diagonals.
Given that the trianglesOAB,OBC,OCD, andODAhave the same perimeter,