Rough automatic draft, low quality: expect mistakes and awkward wording. It will be replaced by a better translation. Original →
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In one of the squares in Krasnodar there is a round flower bed located in the center of the square square ABCD (Fig. 1). Two friends - Vasya and Petya - go to school right through this square. Moreover, sometimes they choose the AMC path (red line), on other days - the ADC path (blue line). Still, it is true that the first path is shorter. Petya believes that the second path is shorter, and here’s why: the circle they have to go around has a rather large radius. “Of course,” says Petya, “if the radius of the circle were not large or if there were no flower beds at all, then the path along the diagonal AC would be clearly shorter than the broken line ADC (Fig. 2). But the size of the circle is considerable, and if we assume that the circle were inscribed in a square, then, Petya argues, we would have a picture like the one in figure 3.” Now Vasya is convinced that the blue line is shorter than the red one: along the “blue” road you have to go “straight” all the time, making only one turn, while along the “red” road you have to make several turns, and sometimes it even seems to go back.
Two friends decided to dispel their doubts in this way: from point A they took different paths: Vasya through the park on the red line, Petya - around the blue line. They approached point C almost simultaneously. The dispute remained unresolved. Then they went back (each their own way) to point A. Now Petya arrived before Vasya. And the dispute was finally resolved by Petya’s crawl. True, it seemed to Vasya that Petya was faster than Vasya.
At home, Vasya decided to do mathematical calculations, denoting the radius of the circle by R, and the side of the square by a.
“It’s strange,” thought Vasya, “the answer to the question of the problem does not depend on the size of the radius of the circle.” However,” he continued his thought, “it was possible to do without calculations: after all, every boundary line, as is known, is longer than any boundary line that has the same ends.
What calculations did Vasya perform and what result did he get when using SEL?