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PG(10)04 - Problems in Geometry(4 1 Consider a quadrangle...

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Problems in Geometry (4) 1. Consider a quadrangle ABCD with points P, Q, R, S, T on BD, AB, AD, CB, CD respectively. Prove that if P, Q, R are collinear and P, S, T are collinear, then the lines QS, RT, AC are concurrent. 1 2. The following is a problem by the late Prof. Demir who liked to present it with a quaintly arithmetic notation which is very suggestive and delightful. A numeral m written decimally will stand for a point and given such numerals m, n the line through them is denoted by m · n : Given distinct non-collinear points 1 , 2 , 3 , 4 consider points 12 1 · 2 - { 1 , 2 } , 23 2 · 3 - { 2 , 3 } , 34 3 · 4 - { 3 , 4 } . Let 1 · 23 and 12 · 3 , 2 · 34 and 23 · 4 meet in 123 , 234 , respectively. Finally, let 1 · 234 intersect 123 · 4 in 1234 . Prove that 12 , 1234 , 34 are collinear. 2 3. Given triangle ABC, let D be a point on BC, and P, Q points on AB. Let PD meet AC in H, QD meet AC in K and CP meet AD in M and CQ meet AD in N. Prove that KM and HN meet on AB.
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