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Assignment 4 - MATH 348 Assignment 4(nal version Leonid...

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MATH 348: Assignment 4 (final version) Leonid Chindelevitch August 2, 2008 1. Two circles, ω 1 and ω 2 , and a line l are given. Locate a line, parallel to l , so that the distance between the points at which this line intersects ω 1 and ω 2 is equal to the length of a given segment AB . 2. Pass a line through a given point A so that the segment included between its point of intersection with a given line l and its point of intersection with a given circle ω is divided in half by the point A . 3. Let O 1 , O 2 , . . . , O n be points in the plane and let A 0 B 0 be an arbitrary segment. Let the segment A i B i be obtained from A i - 1 B i - 1 by a half-turn about O i , for 1 i n . If n is even, show that A 0 A n = B 0 B n . Does this assertion remain true if n is odd? 4. Let two lines l 1 and l 2 , a point A , and an angle α be given. Find a circle with center A such that l 1 and l 2 cut off an arc whose angular measure is equal to α . 5. Let lines l 1 , l 2 , l 3 , meeting at a point, be given, together with a point A on one of these lines. Construct a triangle ABC having the lines l 1 , l 2 , l 3 as angle bisectors. 6. What is the (algebraically) smallest possible value that the power of a
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