A blend of methods of recursion theory and topology: A П 0^1 by Kalantari I., Welch L. PDF

By Kalantari I., Welch L.

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3), which we will come back to in Chapter 23. Consider a triangle ABC and three positive numbers p, q, and r. We would like to know whether there exist points T for which TA : TB : TC = p : q : r . 4) We can show as follows that the locus of the points T for which T B : T C = q : r is a circle, called an Apollonius circle. 2 35 implies that S1 and S2 are fixed points. As T S1 and T S2 are perpendicular to each other, the desired locus is the circle with diameter S1 S2 . 4) are the intersection points of two, and therefore three, Apollonius circles.

It follows that the points denoted above by Q1 and Q2 , which are real in any acute triangle, lie on the Euler line. 13) it follows, as expected, that the Euler line is not well defined in an equilateral triangle. 1 Let l be a line (Fig. 1) that meets the sides of triangle ABC in such a A l B B A C C T Fig. 1. way that the perpendiculars at the intersections points A , B , and C concur at a point T . If we also draw T B and T C, then T C BA and T CB A are cyclic quadrilaterals, which implies that the angles C T B and CT B are equal, respectively, to the angles BA C and B A C.

4) or, in the form of a determinant, x y z x1 y1 z1 = 0 . 5) This leads to the conclusion that in trilinear coordinates, the equation of a line is linear (and homogeneous). 4), we deduce that the equation is not only homogeneous in x, y, and z, but also in x1 , y1 , and z1 , and in x2 , y2 , and z2 . Conversely, a homogeneous linear equation ux+vy+wz = 0 represents a line: it joins the points (0, w, −v) and (−w, 0, u), respectively (v, −u, 0). 2 A second coordinate system is that of the barycentric coordinates (barycenter is another word for center of mass).

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A blend of methods of recursion theory and topology: A П 0^1 tree of shadow points by Kalantari I., Welch L.


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