such that for all 
with incenter 
such that 
Lines 
and 
intersect line 
respectively at 
.
intersects 
at 
and line through 
at 
.
and 
such that 
.
.
such that 
and 
.
.
be a ring not assumed to have an identity, with the following properties:
are nonzero elements of 
.
and not 
be an odd prime number, and let 
denote the field of integers modulo ![$\mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/e/5/a/e5a21492095c9116b100e2f7626a70cc26834e82.png)
be the ring of polynomials over ![$q(x) \in \mathbb{F}_p[x]$](http://latex.artofproblemsolving.com/2/4/e/24e574b86f85cc81ad02b34d6d4ec576a53a95ae.png)
be given by 
where 
mod 
such that 
divides 
in 
be a 
-
and 
,
and 
is defined as ![\[\langle v, w \rangle = \sum_{i=1}^{10} v_i w_i.\]](http://latex.artofproblemsolving.com/9/f/9/9f9b7ff2fb19fee7e935c1b18dea3ec80547b9f9.png)
For 
,
on ![\[ P_u : V \to V, \quad x \mapsto x - \frac{2\langle x, u \rangle u}{\langle u, u \rangle} \]](http://latex.artofproblemsolving.com/9/6/4/9646290e2b5e540b7f85d0d71e4d8992ccd8f542.png)
Given 
satisfying![\[ 0 < \langle v, w \rangle < \sqrt{\langle v, v \rangle \langle w, w \rangle} \]](http://latex.artofproblemsolving.com/d/8/0/d800fe48958cfee9ac01aa8bf11fe431ef45f8ef.png)
let 
.
satisfying 
is 
to 
Y by Adventure10, Mango247
The quadrilateral 
is inscribed in a circle which has diameter 
. Points 
and 
are symmetric to 
and 
with respect to the lines and 
respectively. If the lines 
and intersect at 
, and the lines and 
intersect at 
, prove that 
is perpendicular to .
is inscribed in a circle which has diameter 
. Points 
and 
are symmetric to 
and 
with respect to the lines and 
respectively. If the lines 
and intersect at 
, and the lines and 
intersect at 
, prove that 
is perpendicular to .
.
(i.e. the circumcircle of 
we get
.
thus
.
imply that 
thus 
,
is the internal bisector of 
.
thus 
is the
is harmonic.
is a harmonic quadruple.Thus
.
.
are symmetric to 
and 
with respect to 
, 
,
.
is circumscribed. Than we need to prove that 
.
.
and 
.
and 
.
.
