Calculus: Analytic Geometry and Calculus, with Vectors

24 Analytic geometry in two dimensions

Since the line segment R1M1 is a diameter of the circle, this implies that Q1 must
be on the circle. Cyclic advances of subscripts prove that Q2 and Qa lie on the
circle. This completes the proof of the nine-point-circle theorem.
26 This problem involves the intersection of the medians of the triangle hav-
ing vertices P1(x1,y1), P2(x2,y2), P3(x3,ys) For each k = 1, 2, 3, let Mk be the
mid-point of the side opposite Pk. Show that the equation of the line containing
the median P1M1 can be put in the form

(1) (y2 + Ya - 2y1)(x - x1) - (X2 + X3 - 2x1)(y - y1) = 0.

Show that if we define z and y by the formulas

(2)

then

x1+32+xa Y1+Y2+Y8,

3

x2+x3 -2x1=x1+x2+x3 -3x1=3(x-x1)

y2+ya-2y1=y1+y2+y3-3y1=3(y - y1)

and (1) can be put in the form

(3) (y - y1)(x - x1) - (x - x1)(y - y1) = 0.

Show that (3) implies that the point (x,y) lies on the median P1M1. Finally,

show how this work can be modified to prove that the point (R,9) lies on the other

two medians and hence is the point of intersection of the medians. Remark:

One reason for interest in this matter can be understood when we know enough

about centroids. The point (z,y), the intersection of the medians, is the centroid

of the triangular region bounded by the triangle. It is also the centroid of the

set consisting of the three vertices of the triangle. Moreover, it is the centroid

of the triangle itself, that is, the set consisting of the sides of the triangle. The

coordinates of the intersection of the medians were obtained ina tricky way.

It is possible to put the equation (1) of the median P1M1 and the equation of the

median P2M2 in the forms

(4) (y2 + ya - 2y1)x - (x2 + xa - 2x1)y

= (Y2 + ya - 2y1)x1 - (x2 + xs - 2x1)Y1

(5) (ys + y1 - 2y2)x - (Xs + x1 - 2x2)y

_ (ya + y1 - 2y2)x2 - (xa + x1 - 2x2)Y2

and obtain the coordinates of the intersection of the medians by solving these

equations for x and y without using trickery. There is, however, no guarantee

that time invested in a study of (4) and (5) will produceattractive dividends.

It is easy to obtain ponderous formulas forx and y, but it is not so easy to reduce

the formulas to the right members of the formulas (2).

1.4 Distances, circles, and parabolas

formula

(1.41) d (x2-x1) ++(y2-y1)2

gives the distance d betweentwo points Pa(x1,y1) and P2(x2,y2) in the

plane. To prove (1.41), we notice first that ify2 = y1. then the points P1