Assignment 5

Assignment 5 - MATH 348: Assignment 5 (final version)...

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Unformatted text preview: MATH 348: Assignment 5 (final version) Leonid Chindelevitch 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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