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Unformatted text preview: MATH 348: Assignment 4 (ﬁnal version)
August 2, 2008
1. Two circles, ω1 and ω2 , and a line l are given. Locate a line, parallel to
l, so that the distance between the points at which this line intersects ω1
and ω2 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 O1 , O2 , . . . , On be points in the plane and let A0 B0 be an arbitrary
segment. Let the segment Ai Bi be obtained from Ai−1 Bi−1 by a half-turn
about Oi , for 1 ≤ i ≤ n. If n is even, show that A0 An = B0 Bn . Does this
assertion remain true if n is odd?
4. Let two lines l1 and l2 , a point A, and an angle α be given. Find a circle
with center A such that l1 and l2 cut oﬀ an arc whose angular measure is
equal to α.
5. Let lines l1 , l2 , l3 , meeting at a point, be given, together with a point A
on one of these lines. Construct a triangle ABC having the lines l1 , l2 , l3
as angle bisectors.
6. What is the (algebraically) smallest possible value that the power of a
point can have with respect to a circle of given radius R? What is the
locus of points of constant power with respect to a given circle?
BONUS. Construct an n-gon, given the n points that are the vertices of isoceles triangles constructed on the sides of the n-gon, with the angles α1 , α2 , . . . , αn
at the outer vertices.
BONUS. Prove that the (ordered) convex hull cannot be computed faster than in
Θ(n log n) time, given that n positive real numbers cannot be sorted in
fewer than Θ(n log n) comparisons.
BONUS. In ∆ABC, a = 5, b = 7 and r = 1. Compute c and show that it can be
constructed with marked ruler, but not with ruler and compass.
BONUS. Given lines l, m, n and points P, Q, R, construct ∆ABC with A ∈ l, B ∈
m, C ∈ n, and the extended sides a, b, c pass through P, Q, R, respectively. 1 ...
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