...triangle HMI is, by § 109, right-angled at M, we have, by § 260, MH*; Ml* = HP: PI. But by § 181, Ratio of Similar and Regular Polygons. 266. Theorem....A'B'C' (fig. 109), we have, by § 199, CE~. C'E' = AB: A1B', which, multiplied by the proportion gives I AB X CE -.IjA'B' X C'E'=.4B*: A'B*, and, by § 251,...

...• AD : AB : : AE : AC ; hence (B. H, P. IV.), we have, ADE : ABE : : ABE : ABC ; PROPOSITION XXV. THEOREM. Similar triangles are to each other as the squares of their homologous sides. Let the triangles ABC and DEF be similar, the angle A being equal to the angle D, B to E, and C to...

...equivalent to the sum of the squares El KC and AGHB ; or 28. Corollary. Since To* = TT? + We* iind BC 29. Similar trIangles are to each other as the squares of their homologous sides. Let ABC and D EF be two s similar triangles ; then ABC :DEF—TC2 :DF* Draw BG and E II perpendicular...

...other as the square roots of their areas. This theorem is involved in the theorem that the areas of similar triangles are to each other as the squares of their homologous sides. It is illustrated in the preceding examples. Ex. Construct a triangle with one of its sides 2 in length....

...theorems, and the sum of these areas will be the area of the polygon. PROPOSITION VII.— THEOREM. 20. Similar triangles are to each other as the squares of their homologous sides. Let ABC, A'B'C' be similar tri- <« angles ; then, ABC BC* DC A'B'C' B'C" Let AD, A'D', be the altitudes....

...adjacent sides are 2 and 3, and • then draw a square of the same area. 111. Theorcm.—The areas of similar triangles are to each other as the squares of their homologous sides. ILL. —The meaning of this is, that if ABC and DEF are similar, and any side of ABC is 2 times as...

...right-angled at M, we have, by § 260, MH*: MP = HP:PI. Rut by § 181, Ratio of Similar arid Kegular Polygons. 266. Theorem. Similar triangles are to each...homologous sides. Proof. In the similar triangles ABC, A'B C' (fig. 109), we have, by § 199, CE: C'E'=AB: AB', which, multiplied by the proportion \AB: \A'B'...

...triangle meet in the same point. 5. Problem.—To describe a circle through three given points. 6. Similar triangles are to each other as the squares of their homologous sides. 7. Define: (a) equal figures; (b) equivalent figures; (c) timilar figures. 8. What is the sum of the...

...the squares BIKC and AGHB; or 28. Corollary. AC* = AI? + Btf Since Tc* = Ti? + Fc" THEOREM XIII. 29i Similar triangles are to each other as the squares of their homologous sides. Let ABC and DEF be two B similar triangles; then ABC : DEF = AC 2 : iTF* Draw BG and EH perpendicular...

...the circle. 6. Prove that two regular polygons of the same number of sides are similar. 7. Prove that similar triangles are to each other as the squares of their homologous sides. 8. Show how the area of a polygon circumscribed about a circle may be found; then how the area of a...