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old Vietnamese rhombus

Source: Vietnamese MO (VMO) 1967

August 22, 2018
geometryrhombusoctagonincircleconstruction

Problem Statement

i) ABCDABCD is a rhombus. A tangent to the inscribed circle meets AB,DA,BC,CDAB, DA, BC, CD at M,N,P,QM, N, P, Q respectively. Find a relationship between BMBM and DNDN. ii) ABCDABCD is a rhombus and PP a point inside. The circles through PP with centers A,B,C,DA, B, C, D meet the four sides AB,BC,CD,DAAB, BC, CD, DA in eight points. Find a property of the resulting octagon. Use it to construct a regular octagon. iii) Rotate the figure about the line ACAC to form a solid. State a similar result.
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