MathDB

1

Part of 2021 Iranian Geometry Olympiad

Problems(3)

a shape with at least two reflection symmetries from 4 given shapes

Source: Iranian Geometry Olympiad 2021 IGO Elementary p1

1/25/2022
With putting the four shapes drawn in the following figure together make a shape with at least two reflection symmetries. https://cdn.artofproblemsolving.com/attachments/6/0/8ace983d3d9b5c7f93b03c505430e1d2d189fd.png
Proposed by Mahdi Etesamifard - Iran
geometrysymmetryreflection
IGO 2021 P1

Source: igo 2021 intermediate p1

12/30/2021
Let ABCABC be a triangle with AB=ACAB = AC. Let HH be the orthocenter of ABCABC. Point EE is the midpoint of ACAC and point DD lies on the side BCBC such that 3CD=BC3CD = BC. Prove that BE⊥HDBE \perp HD.
Proposed by Tran Quang Hung - Vietnam
geometry
Proving ∠BHF=90

Source: IGO 2021 Advanced P1

12/30/2021
Acute-angled triangle ABCABC with circumcircle ω\omega is given. Let DD be the midpoint of ACAC, EE be the foot of altitude from AA to BCBC, and FF be the intersection point of ABAB and DEDE. Point HH lies on the arc BCBC of ω\omega (the one that does not contain AA) such that ∠BHE=∠ABC\angle BHE=\angle ABC. Prove that ∠BHF=90∘\angle BHF=90^\circ.
geometrycircumcircleIGO