and circumcenter 
.
the midpoints of 
,
.
with diameter 
,
at a point 
.
meets line 
at 
.
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
Y by
In the triangle 
, the side 
is the smallest. On the sides 
and 
we took the points 
and 
, respectively, such that 
. Let 
be the point of intersection of 
and 
. Let 
be the center of the circle inscribed in the triangle , points and do not coincide. Prove that the line 
is perpendicular to the line .
, the side 
is the smallest. On the sides 
and 
we took the points 
and 
, respectively, such that 
. Let 
be the point of intersection of 
and 
. Let 
be the center of the circle inscribed in the triangle , points and do not coincide. Prove that the line 
is perpendicular to the line .![[asy]
size(7cm);
import geometry;
import olympiad;
pair A,B,C,I,K,L,M;
A=(5,7); B=(-4,-1); C=(7,-1); I=incenter(A,B,C); K=intersectionpoints(circle(A,length(segment(A,C))),A--B)[0]; L=intersectionpoints(circle(C,length(segment(A,C))),B--C)[0]; M=extension(A,L,C,K);
draw(A--B--C--cycle); draw(M--L); draw(M--K); draw(A--M--C); draw(A--extension(A,I,M,C)); draw(C--extension(C,I,M,A)); draw(line(M,I)); draw(L--I--K);
dot(A^^B^^C^^I^^K^^L^^M);
label("$A$",A,NE); label("$B$",B,SW); label("$C$",C,SE); label("$I$",I,NE); label("$K$",K,NW); label("$L$",L,SW); label("$M$",M,S);
[/asy]](http://latex.artofproblemsolving.com/e/0/1/e0175a50776d685b50a4c304a1156edf70e6853a.png)
so 
and 
Hence, 
and 
