27. докажите, что радиус r окружности, вписанной в треугольник, выражается формулой r=2s/a+b+c, где а, b, с - стороны треугольника, s - его площадь. 28.докажите, что радиус r окружности, описанной около треугольника, выражается формулой r=a*b*c/4s, где а, b, с - стороны треугольника, s - его площадь.

1. если соединить центр вписанной окружности с вершинами, то треугольник "разобьется" на три, и в каждом роль высоты будет играть радиус в точку касания. отсюда сразу следует нужная формула s = pr;   p - полупериметр. полезно запомнить её именно в этом виде. важно и то, что такая формула справедлива не только для треугольника, но и для любого выпуклого многоугольника, в который можно вписать окружность. 2. высота к стороне a равна b*sin(c), откуда s = a*b*sin(c)/2; при этом по теореме синусов c = 2*r*sin(c); или sin(c) = c/(2*r); откуда s = a*b*c/4r чтд.

The republic of belarus is situated in the centre of europe. it borders on latvia, lithuania, belarus, the ukrain, russia and poland. la. territory ifs much larger then such countries as austria, belgium, bulgaria, greece and the number of others. its present population is about 12 million people. belarus is a land of vast plains and picturesque hills, thick forests and green meadows, deep blue lakes and flowing rivers. forests cover 2/5 of belarus, they are scattered all over the country. there are a lot of rivers and lakes are in our country. the dnieper, pripyat, beresins, sozh, zapadnaya dvina, neman are the biggest rivers with more then 500 kilometers in length. the largest lake in belarus is lake naroch which stretches 13 kilometers from west to east. it is very popular recreation aria in our country. belarus has a moderately continental climate due to atlantic influences with a worm winter and with a worm summer. to most of the world belarus is relatively known, a new country in the central europe. to its people it's an ancient land. the history of belarus goes back to ancient times beginning with the polotsk principality in the 10th century. from the 13th centuiy till the 16th centuiy the territory of present -day belarus was the core of the grand duchy of lithuania in which belarussian was the official language. it was one of the largest, most powerful and flourishing state*, in eastern europe. the 16th century is considered to be the golden age of belarussion history. in this period there was a wide growth of old and the foundation of many new cities and towns. there occurred significant processes in culture and economy of belarussian people. as belarus is situated in the centre of europe many wars took place in its territory. one of the greatest wars was in 1812 when the emperor napoleon attacked russia. and belarussian lands became the arena, of military actions. a great number of belarussians took part in this war against napoleon. hi the 20th century alone belarus has lived through three revolutions and three bloody wars. the great patriotic war of 1941-1945 is one of the most heroic and tragic periods in the history of the centuiy. for three years the territory of belarus was occupied by the nazi. the country lost more then 3 million people (every third citizen felt in the fight against fascism), many towns and villages were ruined. belarus lost more then half of its national wealth-today belarus is a developed industrial state. industry makes up 70% of the gross national product. the main industries are engineering, chemical industry, wood-working industry, light and food industries. the major industrial centres in belarus are minsk, gomel, vitebsk, mogilev, soligorsk, zhodino, brest and others. belarussian goods are exported to many countries and known, all over the world. agriculture of belarus specializes in milk and meet production, flax, vegetables (especially potatoes) and-fruit (especially apples). the republic of belarus has well-developed science and educational system- the academy of sciences found in 1929 is one of the largest centres of science in our country. its research-institutes work on various important problems in such fields as mathematics, optics, genetics, biology, etc. the demand for higher education remains high. state institutions are the centres both for education and scientific research. nowadays in belarus there are 43 universities, academies and institutes which provide training in various spheres of science and humanities. ai present belarus actively participates in an effective international partnership with the government of different states and international organizations. as so-founder of the united nati6nals belarus contacts with various institutions and organizations of the united nationals, world bank, world health organization, international labor organization, unesco, the european union, etc. belarus cooperates with such countries as germany, denmark, france, canada, italy, netherlands, sweden, great britain, the usa and many others.

Сумма углов параллелограмма и вообще любого четырёхугольника равна 360⁰.  в параллелограмме противоположные углы равны, как и стороны - это его основное свойство. запишем сумму углов параллелограмма: 360⁰ = 2α + 2β, где  α,β - это его углы. у нас сказано, что один из углов больше на 50⁰, то есть  α =  β + 50⁰ . перепишем: 360⁰ = 2(β+50)+2β раскрываем скобки: 360⁰ = 2β + 2β + 100⁰ 360⁰ - 100⁰ = 4β 260⁰ = 4β β = 260⁰ / 4 = 65⁰ теперь находим угол  αα = 50⁰ +  β = 50⁰ + 65⁰ = 115⁰ α = 110⁰проверка: 2β +2α = 2*65⁰ + 2*115⁰ = 130⁰  + 230⁰ = 360⁰   ответ: углы параллелограмма равны 65⁰ и 115⁰