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Unformatted text preview: MATH 348: Assignment 3 (ﬁnal version)
Leonid Chindelevitch
July 22, 2008
1. Show that if a regular ngon can be constructed with a ruler and compass,
then a regular 2ngon can also be constructed with a ruler and compass.
Hint: Use trigonometry and halfangle formulas.
2. Using the doubling method of Archimedes sketched in class, compute a
sequence of approximations to π from above and below by computing the
perimeters of regular polygons inscribed and circumscribed in a circle of
unit diameter starting with a square. Stop when the diﬀerence between
your upper and lower bound is ≤ 10−5 . How many digits of accuracy did
you obtain?
3. Determine for which integers n an angle of n◦ can be constructed with a
ruler and compass.
4. Verify the following geometric solution of the quadratic equation x2 − gx +
h = 0 (with h = 1 if g = 0). The real roots are given by the intersection of
the xaxis and the circle with diameter having endpoints (0, 1) and (g, h).
5. Given compass points that are the vertices of an equilateral triangle, give
a compass construction for the center of the triangle.
6. Suppose 0 < x < π/2. Prove that an angle of x can be constructed with
a ruler iﬀ tan(x) ∈ Q. Hence or otherwise, show that it is impossible to
construct an equilateral triangle with a ruler alone.
BONUS. Construct a square each of whose extended sides passes through one of
four given points.
BONUS. Construct the circles that are tangent to three given circles. [Very hard!] 1 ...
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This note was uploaded on 12/10/2011 for the course MATH 348 taught by Professor Karigiannis during the Summer '06 term at McGill.
 Summer '06
 Karigiannis
 Geometry, Trigonometry

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