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

Source:

December 7, 2021
geometryconstructiongeometric constructioncabri clubs

Problem Statement

level 1
p1. Given an acute triangle ABCABC, point DD in BCBC and point EE in ACAC are marked, so that ∠AEB=∠ADB=90o\angle AEB = \angle ADB = 90^o. The bisectors of the angles ∠CAD\angle CAD and ∠CBE\angle CBE intersect at FF. Find the angle ∠AFB\angle AFB.
p2. [color=#f00](figure missing) Let ABCABC be an equilateral triangle with side 22 and let M,NM, N and PP be the midpoints of ABAB, BCBC and CACA respectively. Draw the three circles of radius 11 centered at A,BA, B, and CC. a) Calculate the area of ​​the shaded region in figure AA. b) The circle that passes through M,NM, N and PP. is drawn. Calculate the area of ​​the shaded region in figure BB.
p3. Let ABCABC be a triangle. Let DD be the midpoint of ABAB and let EE be a point on segment BCBC such that BE=2ECBE = 2 EC. Knowing that ∠BAE=∠ADC\angle BAE = \angle ADC , find the angle ∠BAC\angle BAC.
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