at 
and 
.
meets sides 
and 
again at 
and 
respectively. Let 
be the circumcentre of 
is perpendicular to 
.
for which 
is the square of an integer
of 
if the product of the elements of 
![\[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}\]](http://latex.artofproblemsolving.com/e/7/f/e7fb8c20a43535b5aaed5ab254f1ef043263c62b.png)
true?
.
and 
.
and 
meet at 
.
,
,
,
to all have integer lengths.
Y by Adventure10, Mango247
The right triangles 
and 
are similar and have opposite orientation. The right angles are at 
and 
, and we also have 
. Let 
be the point of intersection of the lines 
and 
. Prove that if the lines 
and 
exist, they are perpendicular.
and 
are similar and have opposite orientation. The right angles are at 
and 
, and we also have 
. Let 
be the point of intersection of the lines 
and 
. Prove that if the lines 
and 
exist, they are perpendicular.
