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18th Cabri Clubs 2007, round 1, level 1, 3 problems, Argentinian geo contest

Source:

December 8, 2021
geometrygeometric constructioncabri clubsconstruction

Problem Statement

level 1
p1. Let PP be a point inside triangle ABCABC such that the line APAP is the bisector of the ∠BAC\angle BAC, BPBP ​​is the bisector of ∠ABC\angle ABC and CPCP is the bisector of ∠ACB\angle ACB. If ∠APB=110o\angle APB = 110^o, ∠BPC=120o\angle BPC = 120^o and ∠CPA=130o\angle CPA = 130^o, find the angles of triangle ABCABC. https://cdn.artofproblemsolving.com/attachments/1/c/6969b4781b5c225a1631a29c2c83754d3d70b4.jpg
p2. Let ABCDABCD be a parallelogram. We mark the midpoint of CDCD, which we call MM. We call P the intersection of BCBC with the line AMAM. It is known that AMAM is the bisector of angle ∠BAD\angle BAD and that PB=10PB = 10. Find the perimeter of parallelogram ABCDABCD.
p3. Let ABCDABCD be a square with side 1 1. Let P,Q,RP, Q, R, and SS be points interior to the square, such that ABPABP, BCQBCQ, CDRCDR, and CASCAS are equilateral triangles. a) Find the area of ​​the quadrilateral PQRSPQRS. b) If all the elements of the figure are deleted except the points P,Q,RP, Q, R and SS, indicate a procedure to reconstruct the square ABCDABCD, using only the ruler and compass.
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