MathDB

4

Part of 2007 Sharygin Geometry Olympiad

Problems(4)

parallelogram with intersections of angle bisectors outside, exist?

Source: 2007 Sharygin Geometry Olympiad Correspondence Round P4

4/25/2019
Does a parallelogram exist such that all pairwise meets of bisectors of its angles are situated outside it?
geometryparallelogramangle bisector
locus of orthocenters

Source: Sharygin Final 2007 8.4

4/30/2019
Determine the locus of orthocenters of triangles, given the midpoint of a side and the feet of the altitudes drawn on two other sides.
geometryLocusorthocenterLocus problems
locus of midpoints of segments, circumcircle related

Source: Sharygin Final 2007 9.4

4/30/2019
Given a triangle ABCABC. An arbitrary point PP is chosen on the circumcircle of triangle ABHABH (HH is the orthocenter of triangle ABCABC). Lines APAP and BPBP meet the opposite sidelines of the triangle at points A′A' and B′B', respectively. Determine the locus of midpoints of segments A′B′A'B'.
geometrycircumcircleLocusLocus problemsmidpoints
symmetric line wrt angle bisectors, reflections of orthocenter, cyclic ABCD

Source: Sharygin 2007 Final 10.4

4/30/2019
A quadrilateral ABCDBCD is inscribed into a circle with center OO. Points C′,D′C', D' are the reflections of the orthocenters of triangles ABDABD and ABCABC at point OO. Lines BDBD and BD′BD' are symmetric with respect to the bisector of angle ABCABC. Prove that lines ACAC and AC′AC' are symmetric with respect to the bisector of angle DABDAB.
geometryreflectionangle bisectorcyclic quadrilateralsymmetry