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1996 Cabri Clubs 2nd, finals, level 1, 4 problems, Argentinian geo contest

Source:

November 25, 2021
geometrycabri clubsconstructiongeometric constructionLocus

Problem Statement

level 1
p1. Three points are given O,GO, G and MM. Construct a triangle in such a way that OO is its circumcenter, GG is its centroid, and MM is the midpoint of one side.
p2. Let ABCABC be a triangle and HH its orthocenter. The height is drawn from AA, which intersects BCBC at DD. On the extension of the altitude ADAD the point EE is taken in such a way that the angles ∠CAD\angle CAD and ∠CBE\angle CBE are equal. Prove that BE=BHBE = BH.
p3. Let ω\omega be a circle, and MM be a variable point on its exterior. From MM the tangents to ω\omega . Let AA and BB be the touchpoints. Find the locus of the incenter of the triangle MABMAB as MM varies.
p4. i) Find a point DD in the interior of a triangle ABCABC such that the areas of the triangles ABDABD, BCDBCD and CADCAD are equal.
ii) The same as i) but with DD outside ABCABC.
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