[/quote]
,
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
and 
points on a plane is 
,
such that: in every set of 
points of a plane, no three of which are collinear, consisting of 
points of every color there exists a 
and 
be isogonal conjugates with respect to 
.
and 
be their respective pedal triangles. Let![\[ X_1=P_2Q_3\cap P_3Q_2,\quad X_2=P_1Q_3\cap P_3Q_1,\quad X_3=P_1Q_2\cap P_2Q_1 \]](http://latex.artofproblemsolving.com/7/f/e/7fe3ec4f69ec074635b2202daad9371e3bb8c72d.png)
Prove that the points 
,
and 
lie on the line 
.
be positive reals strictly smaller than 
,
.
integers are such that 
is prime and 
.
such that 
.
is a circle with diameter 
,
is a circle with diameter 
and 
.
meets 
and 
.
meets 
at 
,
meets 
at 
.
meets 
.
.
, such that 
Prove that :
equality cases : ?
Prove that
![\[\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9\left(ab+bc+ca\right)\]](http://latex.artofproblemsolving.com/e/9/a/e9a57f0700d37fbb28f7d219cc7ac7e52b43acd9.png)
holds for all positive reals 
,
,
.
be a triangle in which 
.
and 
be the incentre and circumcentre of 
be the midpoint of the arc 
.
given that 
.
.
and 
intersect at point 
is symmetric to point 
intersects the circumcircle 
of the triangle 
at the point 
.
is tangent to the circle 
and 
intersect at the point 
and 
.
intersect at point 
.
be the reflection of 
and 
intersect at point 
.
Y by Adventure10, Mango247
A circle is tangential to sides 
and 
of convex quadrilateral 
at 
and 
respectively, and cuts diagonal 
at 
and 
. What are the necessary and sufficient conditions such that there exists another circle which passes through and , and is tangential to 
and 
extended?
and 
of convex quadrilateral 
at 
and 
respectively, and cuts diagonal 
at 
and 
. What are the necessary and sufficient conditions such that there exists another circle which passes through and , and is tangential to 
and 
extended?This post has been edited 1 time. Last edited by orl, Oct 5, 2005, 4:45 PM
tangent to the sides 
.
lie both inside of the segment 
tangent to the lines 
has to be centered the internal bisector of the angle 
.
.
is the radical axis of the circles 
.
,
to the circles 
to the circles 
to the circle 
to the circle 
are also equal. Consequently,
.
