AoPS Academy
on a circle and points 
and 
on segment , such that 
Find the ratio 
, where 
is a point on arc , such that 
is maximized.
let 
and 
be the altitudes. Let the points 
and 
fulfil the following requirements:
is the circumcentre of triangle .
and 
are equal to the circumradius of triangle .
has the same direction as the oriented segment 
. Similarly, 
has the same direction as 
, and 
has the same direction as . is the incentre of triangle .
be a triangle with circumcenter 
. Let be a point on the perpendicular bisector of (see figure) and 
, 
be the intersections of the perpendicular bisectors of 
and 
, respectively, with 
and 
. Prove that the midpoints of 
and and the point are collinear.
be the Miquel point of the cyclic quadrilateral 
It's well-known that 
and 
is parallel to the bisector of 
respectively, so combining 
is the midpoint of arc 
in 
lies on 
and 
(well-known), so the intersection 
of 
lies on the radical axis 
of 
Similarly, the intersection 
of 
resp. lies on 
so notice 
lie on 
we conclude that 
i.e. is the radical axis of 
and 
and two cyclocevian conjugate points 
.
,
are the cevian triangles of 
are the Miquel points of 
;
with respect to 
and 
.