be an acute triangle with 
,
be the projection from 
onto 
be a point on the extension of 
past 
.
be on the perpendicular bisector of 
such that 
are on the same side of ![\[\frac12\angle ALE=1.4\angle ABE+3.4\angle ACE-558^\circ\]](http://latex.artofproblemsolving.com/c/6/3/c63eeb053f1fa8a29bef93516814ae121382219e.png)
Let the reflection of 
be 
and 
,
and 
such that 
.
and 
intersect the circumcircles of 
and 
at 
and 
,
and 
intersect 
and 
at 
and 
.
intersect 
.
.
sastifying 
Y by
Let 
be an arbitrary convex quadrilateral, 
be an arbitrary point of a plane that does not lie on the lines 
, 
, 
, 
. Denote 
the points of intersection of medians of triangles 
,
,
,
respectively. Prove that the area of the quadrilateral 
does not depend on the position of point . Formulate and prove the spatial analogue of the problem.
be an arbitrary convex quadrilateral, 
be an arbitrary point of a plane that does not lie on the lines 
, 
, 
, 
. Denote 
the points of intersection of medians of triangles 
,
,
,
respectively. Prove that the area of the quadrilateral 
does not depend on the position of point . Formulate and prove the spatial analogue of the problem.This post has been edited 2 times. Last edited by parmenides51, May 15, 2021, 6:04 AM
be the midpoints of 
respectively. Since the barycentre divides the medians in a ratio 
,
which maps 
(which is of constant area) to ![$[M_1M_2M_3M_4]=\frac{4}{9}[N_1N_2N_3N_4]$](http://latex.artofproblemsolving.com/e/d/8/ed884fb1caf59a936b7785fab1b04d8bd44473e6.png)
,
,
is the barycentre of the polyhedron corresponding to face 
with vertex ![$[M_1...M_f]$](http://latex.artofproblemsolving.com/3/d/6/3d609afbdd7c7d181a5baa86ab467c85723ca44b.png)
is constant.
of faces 
,
and center ![$[M_1...M_f]=\left(\frac{n}{n+1}\right)^2[N_1...N_f]$](http://latex.artofproblemsolving.com/c/5/a/c5a322cced871a90bd6bb27b2cfb08da06ecbb17.png)
,![$[A...Z]$](http://latex.artofproblemsolving.com/7/4/d/74db5f46750a882a23c53308e5e2259a5da149e5.png)
)
