Докажите, что сумма расстояний от произвольной точки внутри выпуклого равностороннего многоугольника до его сторон величина постоянная.

Ответ в приложенном рисунке.

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1) значит внутренний угол, смежный с углом 135°, равен 45°.треугольник прямоугольный. сумма острых углов равна 90°. значит второй острый угол тоже 45°. треугольник прямоугольный равнобедренный. обозначим один катет х, другой катет х по теореме пифагора х²+х²=10² 2х²=100 х²=50 произведение катетов х·х=х²=50 ответ. 50 2) значит внутренний угол, смежный с углом 120°, равен 60°.треугольник прямоугольный. сумма острых углов равна 90°. значит второй острый угол равен 30°. в прямоугольном треугольнике катет против угла в 30° равен половине гипотенузы. пусть гипотенуза х см, тогда катет против угла в 30°  равен х/2. сумма наибольшей и наименьшей стороны равна х+(х/2)=15 3х/2=15 3х=30 х=10 см гипотенуза 10 см, катет против угла в 30 ° равен 5 см обозначим второй катет у по теореме пифагора y+5²=10² y²+25=100 y²=75 y=5√3 произведение катетов 5·5√3=25√3