be a function such that for every regular 
-
we have 
.
for all reals 
.
)
are the altitudes of the triangle. The bisector of 
intersects 
at 
.
be a point such that 
and 
.
passes through the midpoint of 
& 
.
with 
and at most one of them being zero, show that![\[\frac{1}{\max(a^2,b)}+\frac{1}{\max(b^2,c)}+\frac{1}{\max(c^2,a)}\geq\frac{27}{4}\]](http://latex.artofproblemsolving.com/9/e/1/9e14a9ef2d1a0a98006bcf716fa33c3d9ea29bd5.png)
Y by Adventure10, Mango247
The circle inscribed in the triangle 
touches the sides 
and 
at the points 
and 
, respectively. The angle bisectors from 
and 
intersect the altitude of the triangle from the vertex 
at the points 
and 
, respectively. Prove that one of the points of intersection of the circles circumscribed around the triangles 
and 
lies on 
.
touches the sides 
and 
at the points 
and 
, respectively. The angle bisectors from 
and 
intersect the altitude of the triangle from the vertex 
at the points 
and 
, respectively. Prove that one of the points of intersection of the circles circumscribed around the triangles 
and 
lies on 
.This post has been edited 1 time. Last edited by parmenides51, Mar 25, 2021, 7:25 AM
Reason: vertex instead of vertice
Reason: vertex instead of vertice
be a circle tangent to 
at 
Let 
be the other point of intersection between 
Then 
on 
and 
on 
so 
are collinear as desired. The other statement holds by symmetry.
is tangent to 
since 
and 
will both pass through 
![[asy]
size(10cm);
pair A = dir(115);
pair B = dir(210);
pair C = dir(330);
pair I = incenter(A,B,C);
pair K = foot(I,A,B);
pair L = foot(I,C,A);
pair D = foot(A,B,C);
pair Q = extension(A,D,B,I);
pair R = extension(A,D,C,I);
path c1 = circumcircle(I,R,Q);
path c2 = circumcircle(A,K,L);
pair xi[] = intersectionpoints(c1,c2);
pair X = xi[1];
draw(A--B--C--cycle);
draw(circumcircle(A,K,L),red);
draw(circumcircle(B,K,Q),blue);
draw(circumcircle(C,L,R),blue);
draw(circumcircle(I,Q,R),green);
draw(B--I,green);
draw(C--R,green);
draw(A--D,red+dashed);
dot("$A$",A,dir(A));
dot("$B$",B,dir(B));
dot("$C$",C,dir(C));
dot("$D$",D,dir(D));
dot("$Q$",Q,dir(Q));
dot("$R$",R,dir(R));
dot("$K$",K,dir(K));
dot("$L$",L,dir(L));
dot("$I$",I,dir(I));
dot("$X$",X,dir(X));
[/asy]](http://latex.artofproblemsolving.com/7/1/3/7133824494bbe21e81d5cf00b16e64a5fcd34550.png)
