MathDB

6

Part of 2021 All-Russian Olympiad

Problems(2)

Geometry about a triangle with <B=60

Source: ARO 2021 9.6

4/20/2021
Given is a non-isosceles triangle ABCABC with ∠ABC=60∘\angle ABC=60^{\circ}, and in its interior, a point TT is selected such that ∠ATC=∠BTC=∠BTA=120∘\angle ATC= \angle BTC=\angle BTA=120^{\circ}. Let MM the intersection point of the medians in ABCABC. Let TMTM intersect (ATC)(ATC) at KK. Find TM/MKTM/MK.
geometry
Easy 3d geometry from All-Russian olympiad

Source: All-Russian 2021/11.6

4/20/2021
In tetrahedron ABCSABCS no two edges have equal length. Point A′A' in plane BCSBCS is symmetric to SS with respect to the perpendicular bisector of BCBC. Points B′B' and C′C' are defined analagously. Prove that planes ABC,AB′C′,A′BC′ABC, AB'C', A'BC' abd A′B′CA'B'C share a common point.
geometry3D geometrytetrahedronperpendicular bisector