Problem Set 5. Geometry

Problem Set 5. Geometry - Problem Seminar Fall 2013 Problem...

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Problem Seminar. Fall 2013. Problem Set 5. Geometry. Classical results. 1. Triangle area. Let ABC be a triangle with side lengths a = BC , b = CA , and c = AB , and let r be its inradius and R be its circumradius. Let s = ( a + b + c ) / 2 be its semiperimeter. Then its area is sr = p s ( s - a )( s - b )( s - c ) = abc 4 R = 1 2 ab sin C. 2. Apollonius theorem. Let ABCD be a parallelogram. Then | AB | 2 + | AD | 2 = 1 2 ( | AC | 2 + | BD | 2 ) . Problems. 1. Put 1983. B1. Let v be a vertex (corner) of a cube C with edges of length 4 . Let S be the largest sphere that can be inscribed in C . Let R be the region consisting of all points p between S and C such that p is closer to v that to any other vertex of the cube. Find the volume of R . 2. Put 1986. B1. Inscribe a rectangle of base b and height h and an isosceles triangle of base b in a circle of radius one as shown. For what value of h do the rectangle and triangle have the same area? 3. Put 1982. B1. Let M be the midpoint of side BC of a general 4 ABC . Using the smallest possible n , describe a method for cutting 4 AMB into n triangles which can be

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