be a triangle and 
its circumcentre. A line tangent to the circumcircle of the triangle 
intersects sides 
at 
and 
at 
.
be the image of 
.
is tangent to the circumcircle of triangle 
be a semicircle with diameter 
.
be a point on 
.
touches 
,
in a point 
.
touches 
and 
and 
are perpendicular.
with no parallel sides is inscribed in a circle 
.
are inscribed in triangles 
,
and 
,
,
,
.
Y by NO_SQUARES, buratinogigle, sami1618
The faces of a convex polyhedron are quadrilaterals 
, 
, 
, 
and 
according to the diagram. The edges from points 
and 
, respectively are pairwise perpendicular. Prove that ![\[[ABCD]^2+[ABFE]^2+[ADHE]^2=[BCGF]^2+[CDHG]^2+[EFGH]^2,\]](//latex.artofproblemsolving.com/c/f/b/cfb6dc6726af4711bcfe7827b6531471a75270fb.png)
where ![$[XYZW]$](//latex.artofproblemsolving.com/9/c/2/9c25ef1b5cfb129136334ed811fb699a84f48a13.png)
denotes the area of quadrilateral 
.
Proposed by Géza Kós, Budapest
, 
, 
, 
and 
according to the diagram. The edges from points 
and 
, respectively are pairwise perpendicular. Prove that ![\[[ABCD]^2+[ABFE]^2+[ADHE]^2=[BCGF]^2+[CDHG]^2+[EFGH]^2,\]](http://latex.artofproblemsolving.com/c/f/b/cfb6dc6726af4711bcfe7827b6531471a75270fb.png)
where ![$[XYZW]$](http://latex.artofproblemsolving.com/9/c/2/9c25ef1b5cfb129136334ed811fb699a84f48a13.png)
denotes the area of quadrilateral 
.Proposed by Géza Kós, Budapest
Attachments:
This post has been edited 2 times. Last edited by Miquel-point, Apr 14, 2025, 5:53 PM
then 
by Stokes 
![$a_1 = [ABCD],\ a_2 = [ABFE],\ a_3 = [ADHE],\ g_1 = [BCGF],\ g_2 = [CDHG],\ g_3 = [EFGH]$](http://latex.artofproblemsolving.com/4/7/3/473b49714ce40eee10c4213688ab305608bfb2c3.png)
.
,![\[
a_1\vec{a_1} + a_2\vec{a_2} + a_3\vec{a_3} + g_1\vec{g_1} + g_2\vec{g_2} + g_3\vec{g_3} = \vec{0}.
\]](http://latex.artofproblemsolving.com/2/5/b/25b744c101b2ab90e52a51cf58d8e982517befc5.png)
Rearranging terms, we obtain:![\[
a_1\vec{a_1} + a_2\vec{a_2} + a_3\vec{a_3} = -g_1\vec{g_1} - g_2\vec{g_2} - g_3\vec{g_3}.
\]](http://latex.artofproblemsolving.com/e/6/8/e6898beeceb5fb68f8ffb4877ba7ed13228e383f.png)
Note that the vectors 
are mutually perpendicular, so the dot products between them are zero. The same holds for the vectors 
.![\[
a_1^2 + a_2^2 + a_3^2 = g_1^2 + g_2^2 + g_3^2,
\]](http://latex.artofproblemsolving.com/9/3/b/93b7591b86ee2f15bc042772f795f552ff3e4d21.png)
which is equivalent to:![\[
[ABCD]^2 + [ABFE]^2 + [ADHE]^2 = [BCGF]^2 + [CDHG]^2 + [EFGH]^2.
\]](http://latex.artofproblemsolving.com/6/2/3/623cf6c98877ec0a81ac4a70ab8e9c196c399607.png)
This concludes the proof.
