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1996 Cabri Clubs 2nd Round 1, 6 problems, Argentinian geo contest

Source:

November 25, 2021
geometrygeometric constructioncabri clubsLocusconstruction

Problem Statement

level 1
p1. Construct the given figure, where ABCDABCD is a square and AEFAEF is an equilateral triangle. https://cdn.artofproblemsolving.com/attachments/f/c/1b4043aeed5992ddb8739eec5a8e72ebf4cf91.gif
p2. Let ABCABC be an isosceles triangle (AB=ACAB = AC). We draw the perpendicular bisector mm of ACAC and the bisector nn of angle ∠C\angle C. If m,nm, n and ABAB intersect at a single point, how much is angle ∠A\angle A?
p3. Let A A, B B, and CC be points on a circle. Let us call the orthocenter of the triangle HH. Find the locus of HH as AA moves around the circle.

level 2
p4. Given 33 points A A, B B and CC, construct the isosceles trapezoid ABCDABCD where AB=CDAB = CD and BCBC is parallel to ADAD (BCBC different from ADAD).
p5. Let A A, B B and CC be points on a circle. Let's call the centroid of the triangle GG. Find the locus of GG as AA moves along the circle.
p6. Given a triangle ABCABC, let DD, EE, and FF be the midpoints of the sides BCBC, CACA, and ABAB, respectively. From DD the lines M1M_1 and M2M_2 are drawn, perpendicular on ABAB and ACAC respectively. From EE the lines M3M_3 and M4M_4 are drawn, perpendicular on BCBC and ABAB respectively. From FF the lines M5M_5 and M6M_6 are drawn perpendicular on ACAC and BCBC respectively. Let A′A' be the intersection between M4M_4 and M5M_5. Let B′B' be the intersection between M6M_6 and M1M_1. Let C′C' be the intersection between M2M_2 and M3M_3. Show that the triangles ABCABC and A′B′C′A'B'C' are similar and find the ratio of similarity.
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