99 Points of Intersection: Examples, Pictures, Proofs by Hans Walser

By Hans Walser

The ninety nine issues of intersection provided right here have been gathered in the course of a year-long look for impressive concurrence of traces. for every instance we discover compelling proof for the occasionally startling proven fact that in a geometrical determine 3 instantly traces, or occasionally circles, go through one and an identical element. after all, we're conversant in a few examples of this from easy basic geometry - the intersection of medians, altitudes, attitude bisectors, and perpendicular bisectors of facets of a triangle. right here there are lots of extra examples - a few for figures except triangles, a few the place much more than 3 instantly traces go through a standard point.The major a part of the publication provides ninety nine issues of intersection in basic terms visually. they're constructed in a series of figures, many with no caption or verbal observation. moreover the booklet includes normal concepts on and examples of the issues of intersection, in addition to a few normal equipment of proving their life. a number of the examples proven within the booklet have been encouraged via questions and recommendations made by means of scholars and high-school lecturers. a number of of these examples haven't just a geometrical, but additionally an fascinating aesthetic, aspect.The ebook addresses high-school scholars and scholars on the undergraduate point in addition to their academics, yet will attract someone attracted to geometry

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3 15. A(Ϫ2, 5), B(4, 2) and C(Ϫ1, Ϫ2), D(3, 6) 16. A(2, 0), B(1, Ϫ2) and C(4, 2), D(Ϫ8, 4) x 17. If the line passing through the points (1, a) and (4, Ϫ2) is parallel to the line passing through the points (2, 8) and (Ϫ7, a ϩ 4), what is the value of a? 18. If the line passing through the points (a, 1) and (5, 8) is parallel to the line passing through the points (4, 9) and (a ϩ 2, 1), what is the value of a? –3 y d. 4 19. Find an equation of the horizontal line that passes through (Ϫ4, Ϫ3). 2 20.

What advantage does the Cartesian coordinate system have over the coordinate system under consideration? Comment on your answer. 1 Solutions to Self-Check Exercises 1. a. The points are plotted in the accompanying figure. The distance between A and C is d1A, C 2 ϭ 21Ϫ3 Ϫ 42 2 ϩ 31 Ϫ 1Ϫ22 4 2 y 5 ϭ 21Ϫ72 2 ϩ 32 ϭ 149 ϩ 9 ϭ 158 B(2, 3) c. We will show that C(–3, 1) x –5 5 A(4, –2) [d(A, C)]2 ϭ [d(A, B)]2 ϩ [d(B, C)]2 From part (b), we see that [d(A, B)]2 ϭ 29, [d(B, C)]2 ϭ 29, and [d(A, C )]2 ϭ 58, and the desired result follows.

1 Solutions to Self-Check Exercises 1. a. The points are plotted in the accompanying figure. The distance between A and C is d1A, C 2 ϭ 21Ϫ3 Ϫ 42 2 ϩ 31 Ϫ 1Ϫ22 4 2 y 5 ϭ 21Ϫ72 2 ϩ 32 ϭ 149 ϩ 9 ϭ 158 B(2, 3) c. We will show that C(–3, 1) x –5 5 A(4, –2) [d(A, C)]2 ϭ [d(A, B)]2 ϩ [d(B, C)]2 From part (b), we see that [d(A, B)]2 ϭ 29, [d(B, C)]2 ϭ 29, and [d(A, C )]2 ϭ 58, and the desired result follows. 2. The distance between city A and city B is –5 d1A, B2 ϭ 2200 2 ϩ 50 2 Ϸ 206 b. The distance between A and B is d1A, B2 ϭ 212 Ϫ 42 2 ϩ 3 3 Ϫ 1Ϫ2 2 4 2 ϭ 21Ϫ2 2 2 ϩ 52 ϭ 14 ϩ 25 ϭ 129 The distance between B and C is d1B, C2 ϭ 21Ϫ3 Ϫ 2 2 2 ϩ 11 Ϫ 32 2 ϭ 21Ϫ52 2 ϩ 1Ϫ22 2 ϭ 125 ϩ 4 ϭ 129 or 206 mi.

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