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
An acute-angled triangle 
is given, through the vertices 
and 
of which a circle 
, 
, is drawn. We consider all points 
, that do not lie on none of the lines 
and 
and for which the common tangents of the circumscribed circles of triangles 
and 
are not parallel. Let 
be the point of intersection of such two common tangents.
a) Prove that the locus of points lies to some two lines.
b) Prove that if the circle passes through the orthocenter of the triangle , then one of these lines is the line 
.
is given, through the vertices 
and 
of which a circle 
, 
, is drawn. We consider all points 
, that do not lie on none of the lines 
and 
and for which the common tangents of the circumscribed circles of triangles 
and 
are not parallel. Let 
be the point of intersection of such two common tangents.a) Prove that the locus of points
lies to some two lines.b) Prove that if the circle
passes through the orthocenter of the triangle , then one of these lines is the line 
.This post has been edited 1 time. Last edited by parmenides51, May 7, 2021, 12:43 AM
with power 
and reflect across the angle bisector 
)
and 
be the midpoints of minor and major arc 
.
and 
go to lines 
and 
,
is the reflection of 
we see that 
or 
,
lies on one of the circumferences of diameter 
or 
,
of 
and so 
are the circumcenter and orthocenter respectively of 
,
.
