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19th Cabri Clubs 2007, round 1, level 2, 3 problems, Argentinian geo contest

Source:

December 7, 2021
geometrycabri clubsconstructiongeometric construction

Problem Statement

level 2
p4. Let ABCABC be a triangle with side ABAB less than side ACAC. DD is marked on the side ACAC, such that AD=ABAD = AB. Knowing that ∠ABC−∠ACB=48o\angle ABC - \angle ACB = 48^o, calculate the measure of the angle ∠DBC\angle DBC.
p5. Let ABCABC be a right triangle at AA. D,ED, E, and FF are marked on side BCBC so that ADAD is perpendicular to BCBC, AEAE is the bisector of ∠BAC\angle BAC, and FF is the midpoint of BCBC. Prove that AEAE is a bisector of ∠DAF\angle DAF.
p6. Let ABCDABCD be a parallelogram and EE a point on side BCBC. The line AEAE, which intersects the extension of the side DCDC at FF, and the diagonal BDBD, which intersects the segment AEAE at GG. Knowing that AG=6AG = 6 and GE=4GE = 4, find the length of the drawn segment EFEF.
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