be a triangle 
)
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
Let 
be an acute angled triangle, let 
be its circumcentre, and let 
be points on the sides 
, respectively. The circle 
of radius 
, centered at 
, crosses the segment 
at 
and the circumcircle 
of the triangle again at 
. Similarly, the circle 
of radius 
, centered at 
, crosses the segment 
at 
and the circle again at 
. Finally, the circle 
of radius 
, centered at 
, crosses the segment 
at 
and the circle again at 
. Prove that the quadrilaterals 
and 
are all cyclic, and their circumcircles share a common point.
Evangelos Psychas (Greece)
be an acute angled triangle, let 
be its circumcentre, and let 
be points on the sides 
, respectively. The circle 
of radius 
, centered at 
, crosses the segment 
at 
and the circumcircle 
of the triangle again at 
. Similarly, the circle 
of radius 
, centered at 
, crosses the segment 
at 
and the circle again at 
. Finally, the circle 
of radius 
, centered at 
, crosses the segment 
at 
and the circle again at 
. Prove that the quadrilaterals 
and 
are all cyclic, and their circumcircles share a common point.Evangelos Psychas (Greece)
This post has been edited 1 time. Last edited by parmenides51, Oct 8, 2017, 9:32 AM
Reason: added author
Reason: added author
and
.
are collinear and 
so 
Hence, 
so 
is cyclic. Also, note that 
and
(by 
is cyclic. Since three points on a circle determine a unique circle, then 
is concyclic. Similarly, 
and 
are also concyclic and concur at 
,
