The applications of this theorem are very numerous; for instance, we derive from it Pascal’s theorem of the inscribed hexagon. Consider a hexagon inscribed in a conic. The three alternate sides constitute a cubic, and the other three alternate sides another cubic. The cubics intersect in 9 points, being the 6 vertices of the hexagon, and the 3 Pascalian points, or intersections of the pairs of opposite sides of the hexagon. Drawing a line through two of the Pascalian points, the conic and this line constitute a cubic passing through 8 of the 9 points of intersection, and it therefore passes through the remaining point of intersection—that is, the third Pascalian point; and since obviously this does not lie on the conic, it must lie on the line—that is, we have the theorem that the three Pascalian points (or points of intersection of the pairs of opposite sides) lie on a line.

16. Metrical Theory resumed. Projections and Perpendiculars.—It is a metrical fact of fundamental importance, already used in § 8, that, if a finite line PQ be projected on any other line OO′ by perpendiculars PP′, QQ′ to OO′, the length of the projection P′Q′ is equal to that of PQ multiplied by the cosine of the acute angle between the two lines. Also the algebraical sum of the projections of the sides of any closed polygon upon any line is zero, because as a point goes round the polygon, from any vertex A to A again, the point which is its projection on the line passes from A′ the projection of A to A′ again, i.e. traverses equal distances along the line in positive and negative senses. If we consider the polygon as consisting of two broken lines, each extending from the same initial to the same terminal point, the sum of the projections of the lines which compose the one is equal, in sign and magnitude, to the sum of the projections of the lines composing the other. Observe that the projection on a line of a length perpendicular to the line is zero.

Let us hence find the equation of a straight line such that the perpendicular OD on it from the origin is of length ρ taken as positive, and is inclined to the axis of x at an angle xOD = α, measured counter-clockwise from Ox. Take any point P(x, y) on the line, and construct OM and MP as in fig. 48. The sum of the projections of OM and MP on OD is OD itself; and this gives the equation of the line

x cos α + y sin α = ρ.

Observe that cos α and sin α here are the sin α and −cos α, or the −sin α and cos α of § 8 according to circumstances.

We can write down an expression for the perpendicular distance from this line of any point (x′, y′) which does not lie upon it. If the parallel through (x′, y′) to the line meet OD in E, we have x′ cos α + y′ sin α = OE, and the perpendicular distance required is OD − OE, i.e. ρ − x′ cos α − y′ sin α; it is the perpendicular distance taken positively or negatively according as (x′, y′) lies on the same side of the line as the origin or not.

The general equation Ax + By + C = 0 may be given the form x cos α + y sin α − ρ = 0 by dividing it by √(A² + B³). Thus (Ax′ + By′ + C) ÷ √(A² + B²) is in absolute value the perpendicular distance of (x′, y′) from the line Ax + By + C = 0. Remember, however, that there is an essential ambiguity of sign attached to a square root. The expression found gives the distance taken positively when (x′, y′) is on the origin side of the line, if the sign of C is given to √(A² + B²).

17. Transformation of Coordinates.—We often need to adopt new axes of reference in place of old ones; and the above principle of projections readily expresses the old coordinates of any point in terms of the new.

Fig. 53.

Suppose, for instance, that we want to take for new origin the point O′ of old coordinates OA = h, AO′ = k, and for new axes of X and Y lines through O′ obtained by rotating parallels to the old axes of x and y through an angle θ counter-clockwise. Construct (fig. 53) the old and new coordinates of any point P. Expressing that the projections, first on the old axis of x and secondly on the old axis of y, of OP are equal to the sums of the projections, on those axes respectively, of the parts of the broken line OO′M′P, we obtain: