be inscribed in the circle 
.
intersects 
and 
at points 
and 
,
be the reflection of 
be the reflection of 
of triangle 
lies on the circle 
and 
coincide.
satisfying
Prove
:![\[
1 \cdot 3 \cdot 5 \cdot \ldots \cdot (2n + 1) \mid (4n + 3)(4n + 5) \cdot \ldots \cdot (8n + 3).
\]](http://latex.artofproblemsolving.com/3/1/a/31a0eac3c7ca298c8fa221083dc94e127e705784.png)
and 
.
?
lie on the same plane.
and 
.
Y by Adventure10, Mango247, and 1 other user
Let 
be a regular hexagon with side length 
. At point 
, the perpendicular 
, with length 
, is erected on the hexagon's plane. The points 
and 
are the projections of point on the lines 
and 
, respectively.
be a regular hexagon with side length 
. At point 
, the perpendicular 
, with length 
, is erected on the hexagon's plane. The points 
and 
are the projections of point on the lines 
and 
, respectively.
- Prove that the points
lie on the same plane. - Find the measure of the angle between the planes
and 
.
then 
are images of 
under the inversion with center 
and power 
Thus they lie on the inverse 
of the circumcircle hexagon 
the intersection of the inverse sphere of the plane 
i.e. 
at 
Hence, the angle 
between their planes 
is the angle formed by their diameters 
and 
So, we have a 
right at 
is projection of 
Since 
and 
then 
