with circumcenter 
and orthocenter 
,
be an arbitrary point on the circumcircle of triangle 
and that line 
is not parallel to line 
.
be the point on the circumcircle of triangle 
is perpendicular to 
be the point on line 
such that 
.
and 
be the points on the circumcircle of triangle 
is a diameter, and 
and 
are parallel. Let 
.
and 
are perpendicular.
and 
are perpendicular.
such that![\[(a^2-b^2)(b^2-c^2) = abc.\]](http://latex.artofproblemsolving.com/7/1/8/7188261c40a45f5e15533100432d36abbf515efd.png)
.
that 
is a prime.
sastifying 
Y by Adventure10
In three houses 
and 
, forming a right triangle with the legs 
and 
, live three beetles 
and 
, capable of moving at speeds of 
and 
, respectively. Suppose that you simultaneously release these bugs from point 
and mark the time after which beetles reach their homes. Find on the plane such a point , where is the last time to reach the house a bug would be minimal.
and 
, forming a right triangle with the legs 
and 
, live three beetles 
and 
, capable of moving at speeds of 
and 
, respectively. Suppose that you simultaneously release these bugs from point 
and mark the time after which beetles reach their homes. Find on the plane such a point , where is the last time to reach the house a bug would be minimal.
