MathDB

2

Part of 2007 Sharygin Geometry Olympiad

Problems(4)

each diagonal of a quadrangle divides it into two isosceles triangles

Source: Sharygin 2007 Correspodence p2

4/30/2019
Each diagonal of a quadrangle divides it into two isosceles triangles. Is it true that the quadrangle is a diamond?
geometrydiagonalsquadrilateralisoscelesIsosceles Triangle
right triangle construction given A,C (right) and a point on angle bisector <B

Source: Sharygin Final 2007 8.2

4/30/2019
By straightedge and compass, reconstruct a right triangle ABCABC (∠C=90o\angle C = 90^o), given the vertices A,CA, C and a point on the bisector of angle BB.
geometryangle bisectorconstruction
isosceles trapezoid inscribed an isosceles trapezoid, equal products wanted

Source: Sharygin Final 2007 9.2

4/30/2019
Points EE and FF are chosen on the base side ADAD and the lateral side ABAB of an isosceles trapezoid ABCDABCD, respectively. Quadrilateral CDEFCDEF is an isosceles trapezoid as well. Prove that AE⋅ED=AF⋅FBAE \cdot ED = AF \cdot FB.
geometrytrapezoidisosceles
concurrent lines, altitudes related, from Sharygin 2007

Source: Sharygin 2007 Final 10.2

4/30/2019
Points A′,B′,C′A', B', C' are the feet of the altitudes AA′,BB′AA', BB' and CC′CC' of an acute triangle ABCABC. A circle with center BB and radius BB′BB' meets line A′C′A'C' at points KK and LL (points KK and AA are on the same side of line BB′BB'). Prove that the intersection point of lines AKAK and CLCL belongs to line BOBO (OO is the circumcenter of triangle ABCABC).
geometryaltitudesCircumcenterconcurrentconcurrency