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Handbook of mathematics for engineers and scienteists part 19

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The feet of the perpendiculars drawn from a point Q on the circumcircle to the three sides of the triangle lie on the same straight line called the Simpson line of Q with respect to the triangle (Fig. 3.3h). The circumcenter, the orthocenter, and the centroid lie on a single line, called the Eider fine (Fig. 3.3c). | 94 Analytic Geometry 4.3.2-4. Angle between two straight lines. We consider two straight lines given by the equations y k1 x b1 and y k2x b2 4.3.2.5 where ki tan 1 and k2 tan 2 are the slopes of the respective lines see Fig. 4.17 . The angle a between these lines can be obtained by the formula tan a k2 -k1 . 1 k1k2 4.3.2.6 where k1k2 -1. If k1k2 -1 then a n. Remark. If at least one of the lines is perpendicular to the axis OX then formula 4.3.2.6 does not make sense. In this case the angle between the lines can be calculated by the formula a 2 - 1. 4.3.2.7 The angle a between the two straight lines given by the general equations A1X B1y C1 0 and A2 x B2 y C2 0 4.3.2.8 can be calculated using the expression tan a A1B2 - A2B1 A1A2 B1B2 4.3.2.9 where A1A2 B1 B2 0. If A1A2 B1B2 0 then a n. Remark. If one needs to find the angle between straight lines and the order in which they are considered is not defined then this order can be chosen arbitrarily. Obviously a change in the order results in a change in the sign of the tangent of the angle. 4.3. Straight Lines and Points on Plane 95 4.3.2-5. Point of intersection of straight lines. Suppose that two straight lines are defined by general equations in the form 4.3.2.8 . Consider the system of two first-order algebraic equations 4.3.2.8 A1 x Bi y C1 0 a v r1 o 4.3.2.10 A2 x B2 y C2 0. Each common solution of equations 4.3.2.10 determines a common point of the tow lines. If the determinant of system 4.3.2.10 is not zero i.e. A2 B2l A1B2 - A2B1 0 4.3.2.11 then the system is consistent and has a unique solution hence these straight lines are distinct and nonparallel and meet at the point A x0 y0 where _ B1C2 - B2C1 _ C1 A2 - C2A1 X0 A1B2 - A2B1 y0 A1B2 - A2B1 . 4.3.2.12 Condition 4.3.2.11 is often written as A . A. 4-3-2-13 Example 2. To find the point of intersection of the straight lines y 2x - 1 and y -4x 5 we solve system 4.3.2.10 2x - y - 1 0 -4x - y 5 0 and obtain x 1 y 1. Thus the intersection point has the coordinates