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Trig1 feb 12

Trig1 feb 12 - a b c 5(IMO 1964 Suppose that a b and c are...

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PEM Level C 12 February 2010 Trigonometry: Law Cosines and Sines Theorems: 1. (Law of Cosines) In 4 ABC , the following equations hold: a 2 = b 2 + c 2 - 2 bc cos A b 2 = a 2 + c 2 - 2 ac cos B c 2 = a 2 + b 2 - 2 ab cos C 2. (Extended Law of Sines) Let R be the circumradius of 4 ABC . Then the following relation- ships hold: a sin A = b sin B = c sin C = 2 R Exercises: 1. Derive Heron’s Formula : The area of 4 ABC is given by ( ABC ) = p s ( s - a )( s - b )( s - c ) , where s = 1 2 ( a + b + c ), the semi-perimeter of 4 ABC . 2. Let P be a point inside the equilateral triangle ABC such that PA = 3, PB = 4, and PC = 5. Find the length of one side of 4 ABC . 3. (IMO 1959) Construct a right triangle with given hypotenuse c such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle. 4. Let ABC be a triangle. Prove that 4 ABC is isosceles if and only if a cos B + b cos C + c cos A = 1 2 (
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Unformatted text preview: a + b + c ) . 5. (IMO 1964) Suppose that a , b , and c are the lengths of the sides of a triangle. Prove that a 2 ( b + c-a ) + b 2 ( a + c-b ) + c 2 ( a + b-c ) ≤ 3 abc. 6. Let D be the foot of the bisector of ∠ A of 4 ABC . Prove that BD : DC = BA : AC . 7. Given the value of a and the measures of the angles B and C of 4 ABC , express ( ABC ) in terms of the given quantities. 8. Let ABC be a triangle. Prove that sin A 2 ≤ a b + c 9. (IMO 1996) Let P be a point inside 4 ABC such that ∠ APB-∠ C = ∠ APC-∠ B . Let D and E be the incenters of 4 APB and 4 APC , respectively. Show that the lines AP , BD , and CE are concurrent....
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