Geometry. figure (e.g. multilateral ABCDEF) into the figure A B C D E F is called homothety, or similarity transformation.

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1 ctober 15, 2017 Geoetry. Siilarity an hoothety. Theores an probles. efinition. Two figures are hoothetic with respect to a point, if for each point of one figure there is a corresponing point belonging to the other figure, such that lies on the line at a istance fro point, an vice versa, for each point of the secon figure there is a corresponing point belonging to the first figure, such that lies on the line E F Hoothety E F at a istance fro point. Here the positive nuber is calle the hoothety (or siilarity) coefficient. Hoothetic figures are siilar. The transforation of one figure (e.g. ultilateral EF) into the figure E F is calle hoothety, or siilarity transforation. Thales Theore orollary 1. The corresponing segents (e.g. sies) of the hoothetic figures are parallel. Thales Theore orollary 2. The ratio of the corresponing eleents (e.g. sies) of the hoothetic figures equals. Exercise. What is the ratio of the areas of two siilar (hoothetic) figures? efinition. onsier triangles, or polygons, such that angles of one of the are congruent to the respective angles of the other(s). Sies which are ajacent to the congruent angles are calle hoologous. In triangles, sies opposite to the congruent angles are also hoologous.

2 Generalize Pythagorean Theore 2. Theore 2. For three hoologous segents,, an belonging to the siilar right triangles, an, where is the altitue of the triangle rawn to its hypotenuse, the following hols, l l l Proof. If we square the siilarity relation for the hoologous segents,, where, an are the legs an the hypotenuse of the triangle, we obtain,. Using the property of a proportion, we ay then write, Pythagorean theore for the right triangle,, we ieiately obtain., wherefro, by Theore 1. If three siilar polygons, P, Q an R with areas, an are constructe on legs, an hypotenuse, respectively, of a right triangle, then, Q Proof. The areas of siilar polygons on the sies of a P a b a c right triangle satisfy an, or,. Using the property of a proportion, we R ay then write,, wherefro, using the Pythagorean theore for the right triangle,, we ieiately obtain.

3 Selecte probles on siilar triangles. Proble 1 (hoework proble #4). In the isosceles triangle point ivies the sie into segents such that. If H is the altitue of the triangle an point is the intersection of an, fin the ratio to. 2 Solution. First, let us perfor a suppleentary construction by rawing the segent parallel to,, where point belongs to the sie, an point to an the altitue. Notice the siilar triangles, 1 F E, which iplies,. y Thales H theore,, an, so that., because. Therefore, the sought ratio is,. Proble 2 (hoework proble #5). In a trapezoi with the bases an, segent parallel to the bases,, connects the opposing sies, an. also passes through the intersection point of the iagonals, an, as shown in the Figure. Prove that. Solution. y Thales theore applie to vertical angles an an parallel lines an,. onsequently, M N. Now, applying the sae Thales theore to angles an an parallel lines an, we obtain, an. Hence,, an.

4 The Law of Lever. The Metho of the enter of Mass. rchiees Law of Lever. "Give e a place to stan on, an I will ove the earth." quote by Pappus of lexanria in Synagoge, ook VIII, c. 340 rchiees of Syracuse generally consiere the greatest atheatician of antiquity an one of the greatest of all tie. rchiees anticipate oern calculus an analysis by applying concepts of infinitesials an the etho of exhaustion to erive an rigorously prove a range of geoetrical theores, incluing the area of a circle, the surface area an volue of a sphere, an the area uner a parabola. rchiees of Syracuse orn c. 287 Syracuse, Sicily Magna Graecia ie c. 212 (age aroun 75), Syracuse He was also one of the first to apply atheatics to physical phenoena, founing hyrostatics an statics, incluing an explanation of the principle of the lever. He is creite with esigning innovative achines, such as his screw pup, copoun pulleys, an efensive war achines to protect his native Syracuse fro the Roan invasion. rchiees erives the Law of Lever fro several siple axios (assuptions), which suarize the everyay experience, in a anner siilar to those in Eucliean geoetry. xio 1. Equal weights at equal istances fro the fulcru balance. Equal weights at unequal istance fro the fulcru o not balance, but the weight at the greater istance will tilt its en of the lever own.

5 xio 2. If, when two weights balance, we a soething to one of the weights, they no longer balance. The sie holing the weight we increase goes own. M xio 3. If, when two weights balance, we take soething away fro one of the, they no longer balance. The sie holing the weight we i not change goes own. rchiees then proves the inverse stateents as propositions (theores). Proposition 1. Weights that balance at equal istances fro the fulcru are equal. Proposition 2. Unequal weights at equal istances fro the fulcru o not balance, but the sie holing the heavier weight goes own. Proposition 3. Unequal weights balance at unequal istances fro the fulcru, the heavier weight being at the shorter istance. Proposition 4. If two equal weights have ifferent centers of gravity then the center of gravity of the two together is the ipoint of the line segent joining their centers of gravity. 2

6 Proposition 4 is just a rephrase of the xio 1, where rchiees tacitly introuces the notion of the enter of Gravity (enter of Mass). The way to unerstan the Proposition 4 is to treat the entire weight as if it is locate at a single point, its center of gravity. In other wors, we can picture each weight (ass) as concentrate in a single point, i. e. as a Point Mass. We shall use ters weight an ass interchangeably, assuing that weight is associate with a ass in the hoogeneous gravitation fiel, an therefore is proportional to the ass. The following observation ieiately follows fro the Proposition 4. orollary. If an even nuber of equal weights have their centers of gravity situate along a straight line such that the istances between the consecutive weights are all equal, then the center of gravity of the entire syste is the ipoint of the line segents joining the centers of gravity of the two weights in the ile. M t this point rchiees proves the Law of Lever, first only for coensurate weights. Proposition 5. oensurate weights (asses) balance at istances fro the fulcru, which are inversely proportional to their agnitues,. M=5w =2w =2l 10x1/2w M=5w =5l 4x1/2w =2w 2 5

7 Proof. Let be the greatest coon easure of weights (asses) an,,,. Let us split weight into saller pieces, each of weight, an weight into saller pieces of weight. Let us now split the segent connecting an into congruent saller segents, an also ark such segents on the opposite sie of weight an such segents on the opposite sie of weight. Let us now place all saller weights at the centers of these segents as shown in the Figure. learly, since each of the initial weights was split into an even nuber of equal pieces, which were place syetrically aroun its initial position, the resultant syste of saller weights has the sae center of gravity as the original weight. n the other han, the obtaine syste of weights has the center of gravity in the ile, at a istance of segents fro the position of weight an segents fro the position of weight, as illustrate in the Figure. Therefore,, which proves the Law of Lever for the coensurate weights. The theore for the incoensurate weights is then proven by reucing to contraiction. Theore (Law of Lever). Incoensurate weights (asses) balance at istances fro the fulcru, which are inversely proportional to their agnitues, Proof. Let weights an be place at istances an fro the fulcru, respectively, such that the Law of Lever is satisfie,. ssue that the weights nevertheless o not balance, for exaple, goes own. Reove a sall aount fro weight, turning it into weight, such that it still goes own, but is now coensurate with. Now an are coensurate, an, which eans that shoul rise. This contraicts our assuption, so an ust balance. Note that in the above rchiees iplies a non-trivial fact that a coensurate weight can be foun that iffers fro the given incoensurate weight by an arbitrarily sall aount. This eans that for any irrational nuber there exists a rational nuber, which iffers fro it as little as we want, i. e. that rational nubers are ense.

8 Metho of the enter of Mass (Mass Points). efinition. For two point asses, an at points an, the center of ass lies at a point on the straight line segent such that When fining the center of ass in a syste of point asses, one can replace any pair of asses, an, with a single point ass having the total ass +, place at the center of ass of the pair. The following iportant properties of the enter of Mass follow ieiately. 1. Every syste of finite nuber of point asses has unique center of ass (M). 2. For two point asses, the M belongs to the segent connecting these points; its position is eterine by the rchiees lever rule: the point s ass ties the istance fro it to the M is the sae for both points. 3. The position of the syste s center of ass oes not change if we ove any subset of point asses in the syste to the center of ass of this subset. In other wors, we can replace any nuber of point asses with a single point ass, whose ass equals the su of all these asses an which is positione at their M. eva s Theore Point Masses We select asses,,, an such that the corresponing centers of ass for each pair are at points an respectively Then, + + +