If _P_ is within the circle, then _xx'_ = _yy'_; if _P_ is on the circle, then _x_ and _y_ become 0, and 0 . _x'_ = 0 . _y'_ = 0; if _P_ is at _P__{3}, then _x_ and _y_, having passed through 0, may be considered negative if we wish, although the two negative signs would cancel out in the equation; if _P_ is at _P__{4}, then _y_ = _y'_ and we have _xx'_ = _y_^2, or _x_ : _y_ = _y_ : _x'_, as stated in the proposition. We thus have an excellent example of the Principle of Continuity, and classes are always interested to consider the result of letting _P_ assume various positions. Among the possible cases is the one of two tangents from an external point, and the one where _P_ is at the center of the circle. Students should frequently be questioned as to the meaning of "product of lines." The Greeks always used "rectangle of lines," but it is entirely legitimate to speak of "product of lines," provided we define the expression consistently. Most writers do this, saying that by the product of lines is meant the product of their numerical values, a subject already discussed at the beginning of this chapter. THEOREM. _The square on the bisector of an angle of a triangle is equal to the product of the sides of this angle diminished by the product of the segments made by the bisector upon the third side of the triangle._ This proposition enables us to compute the length of a bisector of a triangle if the lengths of the sides are known. [Illustration] For, in this figure, let _a_ = 3, _b_ = 5, and _c_ = 6. Then [because] _x_ : _y_ = _b_ : _a_, and _y_ = 6 - _x_, we have _x_/(6 - _x_) = 5/3. [therefore] 3_x_ = 30 - 5_x_. [therefore] _x_ = 3 3/4, _y_ = 2 1/4. By the theorem, _z_^2 = _ab_ - _xy_ = 15 - (8 7/16) = 6 9/16. [therefore] _z_ = [sqrt](6 9/16) = 1/4 [sqrt]105 = 2.5+. THEOREM. _In any triangle the product of two sides is equal to the product of the diameter of the circumscribed circle by the altitude upon the third side._ This enables us, after the Pythagorean Theorem has been studied, to compute the length of the diameter of the circumscribed circle in terms of the three sides. [Illustration] For if we designate the sides by _a_, _b_, and _c_, as usual, and let _CD_ = _d_ and _PB_ = _x_, then (_CP_)^2 = _a_^2 - _x_^2 = _b_^2 - (_c_ - _x_)^2. [therefore] _a_^2 - _x_^2 = _b_^2 - _c_^2 + 2_cx_ - _x_^2. [therefore] _x_ = (_a_^2 - _b_^2 + _c_^2) / 2_c_. [therefore] (_CP_)^2 = _a_^2 - ((_a_^2 - _b_^2 + _c_^2) / 2_c_)^2. But _CP_ . _d_ = _ab_. [therefore] _d_ = 2_abc_ / [sqrt](4_a_^2_c_^2 - (_a_^2 - _b_^2 + _c_^2)^2). This is not available at this time, however, because the Pythagorean Theorem has not been proved.