AoPS Academy
are positive real numbers, then
+1 w
be a quadrilateral and let 
be a line. Let intersect the lines 
at points 
respectively. Given that these six points on are in the order 
, show that the circles with diameter 
are coaxal.
, let 
be the least prime that does not divide 
, and define 
to be the product of all primes less than . In particular, 
For having 
, define 
. Consider the sequence 
defined by 
and ![\[ x_{n+1} = \frac{x_n p(x_n)}{q(x_n)} \]](http://latex.artofproblemsolving.com/8/2/5/825be3bbef6a29de69aead03392ca7215093f320.png)
for 
. Find all 
such that 
.
the vertices of these rectangles and 
their centers. There are exactly two possibilities to inscribe in 
namely two consecutive vertices 
on 
or two opposite vertices 
on and cyclically for 
and 
The former case trivially gives the locus of 
as the union of the sidelines 
on 
The application sending 
to the midpoint of 
is an affine homography, even for any fixed directions, not necessarily 
and 
(check the general configuration at Beautiful locus 3), therefore runs on a line. Considering limiting cases 
and 
it follows that this line passes through the midpoint of and the midpoint of the A-altitude. is the union of the 3 sidelines of 
and the 3 lines joining the midpoint of each side with the midpoint of its correponding altitude.
be the tangential triangle of 
is the symmedian point of 
is the foot of the A-altitude and 
are the midpoints of 
resp. 
is the D-median of 
and 
(both perpendicular to BC) 
Hence, if 
cuts 
at 
we get 
but from the complete quadrilateral 
we have 
i.e. passes through 
Similarly the lines joining the midpoints of and 
with midpoints of the B- and C- altitude pass through 
such that:
with all 
is real number
be an acute triangle.
