,
,
be non-negative real numbers, no two of which are zero. Prove that :
with 
.
is right-angled isosceles.
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the 
cars from the 
Y by Adventure10, Mango247
Let 
be points on the sides 
of a triangle 
, respectively such that the lines 
are concurrent at the point 
. Let a line 
through 
intersect the rays 
and 
at the points 
and 
, respectively. Let 
and 
be points on the rays 
and 
, respectively such that the equation
![\[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \]](//latex.artofproblemsolving.com/8/1/d/81d90e1fec1c1cf100a1b9b20e3148eeecdccf95.png)
holds. Show that the lines
and 
are perpendicular to each other.
be points on the sides 
of a triangle 
, respectively such that the lines 
are concurrent at the point 
. Let a line 
through 
intersect the rays 
and 
at the points 
and 
, respectively. Let 
and 
be points on the rays 
and 
, respectively such that the equation![\[ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} \]](http://latex.artofproblemsolving.com/8/1/d/81d90e1fec1c1cf100a1b9b20e3148eeecdccf95.png)
holds. Show that the lines
and 
are perpendicular to each other.
and find the equality case.
,
and 
,
Also, note that if 
are the projections of 
on 
,
or 
is acute or not,![\[
DQ\cos \angle QDN+DR\cos \angle RDM=|\overrightarrow{DU}-\overrightarrow{DV}|=UV.
\]](http://latex.artofproblemsolving.com/1/b/e/1be471da7f4f6846250662168ef9f26444d6a271.png)
It follows that
where we used Titu's lemma, 
and the trivial estimate 
.
Combining our two estimates, we find that 
,
,
(i.e., 
and 
are parallel), (3) 
and 
.
,
and 
are parallelograms, and thus 
.
,
.
.
is the internal bisector of 
,
