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 
.
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)
be positive reals strictly smaller than 
,
.
Y by Adventure10
Let 
be an acute triangle. Denote by 
and 
the feet of the altitudes from vertices 
and 
, respectively. Let 
be a point inside the triangle such that the line 
is tangent to the circumcircle of the triangle 
and the line 
is tangent to the circumcircle of the triangle 
. Show that the line 
is perpendicular to 
.
be an acute triangle. Denote by 
and 
the feet of the altitudes from vertices 
and 
, respectively. Let 
be a point inside the triangle such that the line 
is tangent to the circumcircle of the triangle 
and the line 
is tangent to the circumcircle of the triangle 
. Show that the line 
is perpendicular to 
.
the third altitude, 
.
; similarly, 
,
and 
hence 
or 
.
，
，
，
,
.
and 
.
.
.
.![[asy]defaultpen(linewidth(0.7));
pair A=origin, B=(12,0), C=(12,5), D=(0,5), P=(6,2.5);
draw(A--B--C--D--A--C^^B--D);
label("$A$", A, SW);
label("$B$", B, SE);
label("$C$", C, NE);
label("$D$",D, NW);
label("$P$", P, N);[/asy]](http://latex.artofproblemsolving.com/6/8/9/68954d502c5415a2c5323fdac045f34b83c94b7a.png)
![[asy]defaultpen(linewidth(0.7));
pair P=origin, B=(8,0), A=(0,3), C=(0,-14), D=(-13,0);
draw(A--B--C--D--A--C^^B--D);
label("$A$", A, N);
label("$B$", B, E);
label("$C$", C,S);
label("$D$",D, W);
label("$P$", P, NE, fontsize(10));
markscalefactor=0.1;
draw(rightanglemark(A,P,D));[/asy]](http://latex.artofproblemsolving.com/b/c/0/bc052d430e6830f58d40f44c04fcfe9a611c705a.png)
<=> 
<=> 
and with the right angle we get the two similar triangles.Where's the mistake?(if 
and 
are above 
and 
Carnot theorem.
altitude.
.
But this is clearly true as 
.
