2016 AIME II Problems/Problem 14
Equilateral has side length
. Points
and
lie outside the plane of
and are on opposite sides of the plane. Furthermore,
, and
, and the planes of
and
form a
dihedral angle (the angle between the two planes). There is a point
whose distance from each of
and
is
. Find
.
Solution
The inradius of is
and the circumradius is
. Now, consider the line perpendicular to plane
through the circumcenter of
. Note that
must lie on that line to be equidistant from each of the triangle's vertices. Also, note that since
are collinear, and
, we must have
is the midpoint of
. Now, Let
be the circumcenter of
, and
be the foot of the altitude from
to
. We must have
. Setting
and
, assuming WLOG
, we must have $(\tan(120^{\circ})=-\sqrt{3}=\dfrac{\dfrac{x+y}{100\sqrt{3}}}{\dfrac{30000-xy}{30000}$ (Error compiling LaTeX. Unknown error_msg). Therefore, we must have
. Also, we must have
by the Pythagorean theorem, so we have
, so substituting into the other equation we have
, or
. Since we want
, the desired answer is
.