By Hopf H.

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Extra info for A New Proof of the Lefschetz Formula on Invariant Points

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B CONSTRUCTION Bisect a Segment Step 1 Draw a segment and −− name it XY. Place the compass at point X. Adjust the compass so that its width is 1 greater than _XY. 2 Draw arcs above and −− below XY. Step 2 Using the same compass setting, place the compass at point Y and draw arcs above −− and below XY that intersect the two arcs previously drawn. Label the points of intersection as P and Q. Step 3 Use a straightedge to draw PQ . Label the point where it −− intersects XY as M. Point M is the midpoint −− of XY, and PQ is a −− bisector of XY.

Congruence marks are used to indicate this. 23. 8 mm C 3 in. 4 1 in. 2 24. QR Z Y 26. 2 cm S R 27. 0 cm 4 Q 5 in. 16 X 25. ST 2 1 in. P W A X B Y C D 3 3 in. 4 Find the value of the variable and ST if S is between R and T. 28. RS = 7a, ST = 12a, RT = 76 29. RS = 12, ST = 2x, RT = 34 30. RS = 2x, ST = 3x, RT = 25 31. RS = 16, ST = 2x, RT = 5x + 10 32. RS = 3y + 1, ST = 2y, RT = 21 33. RS = 4y - 1, ST = 2y - 1, RT = 5y Use the figures to determine whether each pair of segments is congruent. −− −−− −− −−− −− −− 34.

22. y y J (Ϫ3, 4) M (4, 0) O x O K (2, Ϫ4) x L (Ϫ2, Ϫ3) Use the Distance Formula to find the distance between each pair of points. 24. L(3, 5), M(7, 9) 23. J(0, 0), K(12, 9) 25. 26. y y V (5, 7) T (6, 5) U (2, 3) S (Ϫ3, 2) O 26 Chapter 1 Tools of Geometry x x O 27. 28. y y P(3, 4) R(1, 5) Q (–5, 3) x O x O N(–2, –2) Use the number line to find the coordinate of the midpoint of each segment. A B C D E F Ϫ7 Ϫ6 Ϫ5 Ϫ4 Ϫ3 Ϫ2 Ϫ1 0 1 2 3 4 5 6 7 8 9 −− 29. AC −− 32. BD −− 31. CE −− 34. BE −− 30. DF −− 33.

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A New Proof of the Lefschetz Formula on Invariant Points by Hopf H.


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