[/quote]
,
is the midpoint of 
,
the point of intersection of the tangents at points 
and 
of the circumscribed circle of 
is the midpoint of the segment 
.
meets the circumcircle 
.
.
:
be reals such that 
and 
Prove that
Where 
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
,![\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}\geq 4+\frac{8(a-c)^2}{(a+b+c+d)^2}.\]](http://latex.artofproblemsolving.com/7/b/6/7b6e32d32d89ca058a06974e7276f0a5874d90a7.png)
such that 
that satisfy the equation
.
.
and 
,
and 
be the incenter of triangles 
and 
,
.
,
,
,
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
we have 
.
.
Y by
On the sides 
and 
of the triangle 
mark the points 
and 
, respectively, so that 
. The sides of the triangle 
lie on the perpendicular bisectors of
, 
and 
, and the sides of triangle 
are the perpendicular bisectors to 
,
and 
. Find the ratio of the areas 
.
(W. Bryman)
and 
of the triangle 
mark the points 
and 
, respectively, so that 
. The sides of the triangle 
lie on the perpendicular bisectors of
, 
and 
, and the sides of triangle 
are the perpendicular bisectors to 
,
and 
. Find the ratio of the areas 
.(W. Bryman)
This post has been edited 1 time. Last edited by parmenides51, Jun 23, 2021, 7:55 PM
Reason: source edit
Reason: source edit
is uniquely determined as intersection of two perpendicular bisectors
where![$X_1\equiv [AD]\cap [BE]$](http://latex.artofproblemsolving.com/2/6/c/26cade1202b40f42c5fe1c16bdf707bc58ad646a.png)
, ![$Y_1 \equiv [AD] \cap [CF]$](http://latex.artofproblemsolving.com/f/e/2/fe2518370196846061e2396ba0ab115889c9718b.png)
,![$Z_1 \equiv [BE] \cap [CF]$](http://latex.artofproblemsolving.com/e/a/d/eadd7564901457b2ce75241f906748ac06868d1b.png)
![$[XY]$](http://latex.artofproblemsolving.com/b/d/5/bd5db5e85aa6daea3eebecaea5d26721edd15203.png)
denotes perpendicular bisector of segment 
