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# PG(10)02 - Problems in Geometry(2 1 Given a rectangle ABCD...

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Problems in Geometry (2) 1. Given a rectangle ABCD prove that | PA | 2 + | PC | 2 = | PB | 2 + | PD | 2 for any point P. 2. Given an acute angled positively oriented triangle ABC, let O and H be the circum- center and the orthocenter of ABC, respectively. Let AH intersect BC in ˜ A. (A) Prove that | AH | = 2 R cos A. 1 (B) Prove that | ˜ AH | = 2 R cos B cos C . 2 (C) Prove that OH is perpendicular to AH iff tan B tan C = 3 . 3 3. Let P be a point on the circumcircle of a triangle ABC such that AP is the internal bisector of the angle A . Prove that the following conditions are equivalent 4 5 (i) | PA | = 2 | PB | , (ii) 2 sin A 2 = cos B - C 2 , (iii) 2 a = b + c . 4. In a triangle ABC prove that the following assertions are equivalent: (i) A = 90 , (ii) r + r b + r c = r a , (iii) r b r c = rr a . 6 5. Generalise the previous result by proving that in a triangle ABC the following condi- tions are equivalent: (i) cos A = α - 1 α + 1 , (ii) r b + r c = α ( r a - r ) , (iii) r b r c = αrr a . where α 6 = - 1 is a constant. Compute the angle A in a triangle ABC in which 3 r + r b + r c = 3 r a .

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PG(10)02 - Problems in Geometry(2 1 Given a rectangle ABCD...

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