The population of China is one billion people. It would seem that on a map of China with a scale of 1 : 1,000,000 (1 cm : 10 km), it would be possible to fit a million times fewer people than there is in the whole country. However, in fact, not only 1000, but even 100 people will not be able to be placed on this map. Can you explain this contradiction?
In the triangle \(ABC\) the sides are compared as following: \(AC>BC>AB\). Prove that the angles are compared as follows: \(\angle B > \angle A > \angle C\).
Consider a quadrilateral \(ABCD\). Choose a point \(E\) on side \(AB\). A line parallel to the diagonal \(AC\) is drawn through \(E\) and meets \(BC\) at \(F\). Then a line parallel to the other diagonal \(BD\) is drawn through \(F\) and meets \(CD\) at \(G\). And then a line parallel to the first diagonal \(AC\) is drawn through \(G\) and meets \(DA\) at \(H\). Prove the \(EH\) is parallel to the diagonal \(BD\).
Cut an arbitrary triangle into parts that can be used to build a triangle that is symmetrical to the original triangle with respect to some straight line (the pieces cannot be inverted, they can only be rotated on the plane).