648 Geometry and Trigonometry
2 S 1 + 2 S 2 + 2 S 3 ≥
1
2
(A(ABC)+S 1 +S 2 +S 3 ).
The inequality follows.
(M. Pimsner, S. Popa,Probleme de geometrie elementara ̆(Problems in elementary
geometry), Editura Didactica ̧ ̆si Pedagogica, Bucharest, 1979) ̆
647.Assume that the two squares do not overlap. Then at most one of them contains
the center of the circle. Take the other square. The line of support of one of its sides
separates it from the center of the circle. Looking at the diameter parallel to this line, we
see that the square is entirely contained in a half-circle, in such a way that one of its sides
is parallel to the diameter. Translate the square to bring that side onto the diameter, then
translate it further so that the center of the circle is the middle of the side (see Figure 90).
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Figure 90The square now lies inside another square with two vertices on the diameter and
two vertices on the circle. From the Pythagorean theorem compute the side of the larger
square to be
√
4
5. This is smaller than 0.9, a contradiction. Therefore, the original squares
overlap.
(R. Gelca)
648.The Möbius band crosses itself if the generating segments at two antipodal points
of the unit circle intersect. Let us analyze when this can happen. We refer everything
to Figure 91. By construction, the generating segments at the antipodal pointsMandN
are perpendicular. LetPbe the intersection of their lines of support. Then the triangle
MNPis right, and its acute angles areα 2 andπ 2 −α 2. The generating segments intersect if
they are longer than twice the longest leg of this triangle. The longest leg of this triangle
attains its shortest length when the triangle is isosceles, in which case its length is
√
2.
We conclude that the maximal length that the generating segment of the Möbius band
can have so that the band does not cross itself is 2
√
2.
649.Comparing the perimeters ofAOBandBOC, we find that‖AB‖+‖AO‖=
‖CB‖+‖CO‖, and henceAandCbelong to an ellipse with fociBandO. The same
argument applied to trianglesAODandCODshows thatAandCbelong to an ellipse
with fociDandO. The foci of the two ellipses are on the lineBC; hence the ellipses are
symmetric with respect to this line. It follows thatAandCare symmetric with respect