...which each of these excellent works takes with Euclid's Prop, ig, Bk. vi.- — " similar triangles are to one another in the duplicate ratio of their homologous sides " — mysterious but high-sounding- to countless generations of schoolboys. Here it is, in identical...

...quantity of land by a line parallel to any one of its sides. RULE. — The areas of similar triangles are to one another in the duplicate ratio of their homologous sides ; hence, as the area of the triangle ABC is to the square of the side AC, or BC, so is the area of...

...without the circle, is equal to the square of the line which touches it. 6. Prove that similar triangles are to one another in the duplicate ratio of their homologous sides. Given (b) the base of a triangle, find an expression for the base of a similar triangle whose area...

...rectangle contained by the parts. 2. Deseribe a regular pentagon about a given cirele. 3. Similar triangles are to one another in the duplicate ratio of their homologous sides. 4. If perpendiculars Aa, B&, Cc, be drawn from the angular points of a triangle ALC upon the opposite...

...described, on a given line, similar to a given fig. QEF PROPOSITION XIX. THEOREM. Similar triangles are to one another in the duplicate ratio of their homologous sides. Let ABC, DEF be similar AS, having L s at A, B, C= s.sa.tD,E,F respectively, so that BC and EF are...

...AEDCB) may be divided inl» similar triangles, equal in number, and homologous to all. Ana the polygons are to one another in the duplicate ratio of their homologous sides. PART 1. — Because in the triangles FGI and AED, the angles G and E are ' equal, and the sides about...

...stand. cal angle and the segments into which the line bisecting it divides the base. 4. Similar polygons are to one another in the duplicate ratio of their homologous sides. 5. The rectangle contained by the diagonals of a quadrilateral inscribed in a circle is equal to the...

...duplicate ratio of their homologous sides ; and it has already been proved in triangles (VI. 19) ; therefore, universally, similar rectilineal figures...in the duplicate ratio of their homologous sides. COT. 2. And if to AB, FG, two of the homologous sides, a third proportional M be taken (VI. 11), 1....

...equal, then the greatest area is possessed by the figure which has the largest number of sides. (7) Similar rectilineal figures are to one another in the duplicate* ratio of their homologous^ sides. (8) If three straight lines oe proportionals, as the first quantity is to the third quantity, so is...

...duplicate ratio, and illustrate its meaning as you would to a class. (b.) Prove that similar triangles are to one another in the duplicate ratio of their homologous sides. 2. Describe a circle which will pass through a given point, and touch a given circle in a given point....