or (_a_ - _b_)/(_b_ - _c_) = _ab_/_bc_ = _a_/_c_. This is the reason why the line _AB_ is said to be divided harmonically. The line _P__{1}_P__{2} is also called the _harmonic mean_ between _AP__{2} and _P__{2}_B_, and the points _A_, _P__{1}, _B_, _P__{2} are said to form an _harmonic range_. [Illustration] It may be noted that [L]_P__{2}_CP__{1}, being made up of halves of two supplementary angles, is a right angle. Furthermore, if the ratio _CA_ : _CB_ is given, and _AB_ is given, then _P__{1} and _P__{2} are both fixed. Hence _C_ must lie on a semicircle with _P__{1}_P__{2} as a diameter, and therefore the locus of a point such that its distances from two given points are in a given ratio is a circle. This fact, Pappus tells us, was known to Apollonius. At this point it is customary to define similar polygons as such as have their corresponding angles equal and their corresponding sides proportional. Aristotle gave substantially this definition, saying that such figures have "their sides proportional and their angles equal." Euclid improved upon this by saying that they must "have their angles severally equal and the sides about the equal angles proportional." Our present phraseology seems clearer. Instead of "corresponding angles" we may say "homologous angles," but there seems to be no reason for using the less familiar word. [Illustration] [Illustration] [Illustration] It is more general to proceed by first considering similar figures instead of similar polygons, thus including the most obviously similar of all figures,--two circles; but such a procedure is felt to be too difficult by many teachers. By this plan we first define similar sets of points, _A__{1}, _A__{2}, _A__{3}, ..., and _B__{1}, _B__{2}, _B__{3}, ..., as such that _A__{1}_A__{2}, _B__{1}_B__{2}, _C__{1}_C__{2}, ... are concurrent in _O_, and _A__{1}_O_ : _A__{2}_O_ = _B__{1}_O_ : _B__{2}_O_ = _C__{1}_O_ : _C__{2}_O_ = ... Here the constant ratio _A__{1}_O_ : _A__{2}_O_ is called the _ratio of similitude_, and _O_ is called the _center of similitude_. Having defined similar sets of points, we then define similar figures as those figures whose points form similar sets. Then the two circles, the four triangles, and the three quadrilaterals respectively are similar figures. If the ratio of similitude is 1, the similar figures become symmetric figures, and they are therefore congruent. All of the propositions relating to similar figures can be proved from this definition, but it is customary to use the Greek one instead. [Illustration]