Unformatted text preview: MATH 348: Assignment 5 (ﬁnal version)
July 29, 2008
1. Suppose that A, B, C, D are 4 distinct points in the plane, with AB ⊥ CD
and AC ⊥ BD. Prove that AD ⊥ BC . Hint: Consider several cases!
2. In an isoceles triangle, prove that the sum of the distances from a point
P on the base to the sides is a constant independent of the position of P .
3. An equilateral triangle ABC is rotated by π/3 about its center to give
A B C . Show that the hexagon common to ABC and A B C is regular.
4. If a circle of radius r is inscribed into a sector (part enclosed by two radii
and the arc they cut oﬀ) of a circle of radius R, ﬁnd the area of the sector.
5. The cyclic quadrilateral KLM N is inscribed in a circle ω . The tangents to
ω drawn through K, L, M, N form a new quadrilateral, also cyclic. Find
the area of KLM N given that it has perimeter p and N M = 2M L = 8LK .
6. The centers of three circles lie on a straight line, and the center of a fourth
circle is at a distance d from this line. Each of these circles is tangent to
the three others. Find the radius of the fourth circle.
7. Given a line l and two points A, B on the same side of l, construct the
circle passing through A and B and tangent to l.
8. Given a circle ω and a line l with point A, construct a circle tangent to ω
and tangent to l at A.
9. Given three pairwise intersecting non-concurrent lines, ﬁnd the locus of
the circumcenters of all possible triangles with one vertex on each line.
10. Given a circle ω and an angle α, ﬁnd the locus of all points P such that
the two tangents from P to ω form an angle α.
BONUS. Consider a parallelogram ABCD, with AB = 1, BC = 2 and an obtuse
angle ∠ABC . Through each of B and D we draw two lines, one of which
is perpendicular to AB , the other, to BC . The intersection of these four
lines forms a parallelogram, similar to ABCD. Find the area of ABCD.
BONUS. In ∆ABC , b = c = 5 and a = 6. Point D lies on AC and P is the point on
BD so that ∠AP C = π/2. If ∠ABP = ∠BCP , ﬁnd the ratio AD : DC . 1 ...
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