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such that 
, for all 
.
is a set of positive integers.)
touch sides 
, 
, and 
at points 
, 
, and 
, respectively. Let 
be the diametrically opposite point of with respect to the incircle. Let lines 
and 
intersect the incircle again at 
and 
, respectively. Prove that the lines 
, 
, and 
are concurrent, i.e., the lines intersect at the same point. is on side of triangle 
. Let 
, 
and 
be nine-point centers of triangles 
, 
and respectively. Prove that perpendicular lines from to 
, from to , from to and from 
to are concurrent.
of real numbers that satisfy the system of equations![\[ \begin{cases}x^3 = 3x-12y+50, \\ y^3 = 12y+3z-2, \\ z^3 = 27z + 27x. \end{cases}\]](http://latex.artofproblemsolving.com/d/4/d/d4d6c485b4653577e732ff235448bb779914a576.png)
let 
and 
be altitudes. Let 
intersect the circumcircle of at 
and 
. Let 
be the reflection of in , and let 
be the reflection of 
in . Prove that 
, , are concyclic.
let and be altitudes. Let intersect the circumcircle of at and . Let be the reflection of in , and let be the reflection of in . Prove that , , are concyclic.
is cyclic. let 
intersects 
at 
. By POP, we have![\[ A'H_A \cdot H_A H = AH_A \cdot H_A A_1 = QH_A \cdot H_A P \]](http://latex.artofproblemsolving.com/4/f/0/4f0e9726a72d661b649bab9e01029369c386f2fc.png)
Hence, 
is cyclic. Do this similarly, and we get 
is cyclic, and we are done.
let and be altitudes. Let intersect the circumcircle of at and . Let be the reflection of in , and let be the reflection of in . Prove that , , are concyclic.
lies on 
, then power of a point will finish it
and intersection .
- cyclic.Similarity 
cyclic.
- cyclic quadrateral.
.
is just the reflection of 
and similarly, 
.
and so 
is cyclic