MathDB

concurrency lines _|_ plane ABC at Lemoine point, by equilateral triangles

Source: 239 MO 2021 10-11 p4

May 2, 2021

concurrentconcurrencyLemoine pointgeometryperpendicular3D geometry

Problem Statement

Symedians of an acute-angled non-isosceles triangle

ABC

intersect at a point at point

L

, and

AA_1

BB_1

and

CC_1

are its altitudes. Prove that you can construct equilateral triangles

A_1B_1C'

B_1C_1A'

and

C_1A_1B'

not lying in the plane

ABC

, so that lines

AA' , BB'

and

CC'

and also perpendicular to the plane

ABC

at point

L

intersected at one point.