Art of Problem Solving { Computational...
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Art of Problem Solving – Computational geometry Proposition (Area formula). Area K of 4ABC can be computed as (with standard notation) K = 1 2 bc sin α = 1 2 ca sin β = 1 2 ab sin γ. Proposition (Law of Cosines). In 4ABC (with standard notation) we have c 2 = a 2 + b 2 - 2ab cos γ. Problem 1 (AIME 2011). On square ABCD, point E lies on the side AD and point F lies on the side BC , so that BE = EF = FD = 30. Find the area of square ABCD. Problem 2. Circle ω 1 is the largest circle which can be inscribed in a semicircle ω with radius 2. Circle ω 2 is tangent to ω internally and to ω 1 externally. Find the radius of ω 2 . Problem 3. In 4ABC let m a be the length of the A-median. Prove that m 2 a = b 2 + c 2 2 - a 2 4 . Problem 4. Squares ABED, BCGF , CAIH are erected externally from the sides of triangle ABC . Show that triangles AID, BEF , and CGH have equal area. Problem 5. Let ω be the angle between diagonals of a convex quadrilateral ABCD. If K is the area of ABCD, prove that K = 1 2 AC · BD · sin ω. Problem 6 (AIME 2005). In quadrilateral ABCD let BC = 8, CD = 12, AD = 10, and ∠A = ∠B = 60 ◦ . Find the distance AB. 1
Transcript of Art of Problem Solving { Computational...
Art of Problem Solving – Computational geometry
Proposition (Area formula). Area K of 4ABC can be computed as (with standard notation)
K =1
2bc sinα =
1
2ca sin β =
1
2ab sin γ.
Proposition (Law of Cosines). In 4ABC (with standard notation) we have
c2 = a2 + b2 − 2ab cos γ.
Problem 1 (AIME 2011). On square ABCD, point E lies on the side AD and point F lies on
the side BC, so that BE = EF = FD = 30. Find the area of square ABCD.
Problem 2. Circle ω1 is the largest circle which can be inscribed in a semicircle ω with radius 2.
Circle ω2 is tangent to ω internally and to ω1 externally. Find the radius of ω2.
Problem 3. In 4ABC let ma be the length of the A-median. Prove that
m2a =
b2 + c2
2− a2
4.
Problem 4. Squares ABED, BCGF , CAIH are erected externally from the sides of triangle
ABC. Show that triangles AID, BEF , and CGH have equal area.
Problem 5. Let ω be the angle between diagonals of a convex quadrilateral ABCD. If K is the
area of ABCD, prove that
K =1
2AC ·BD · sinω.
Problem 6 (AIME 2005). In quadrilateral ABCD let BC = 8, CD = 12, AD = 10, and
∠A = ∠B = 60◦. Find the distance AB.
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