1979 USAMO Problems/Problem 2
Contents
Problem
is the north pole.
and
are points on a great circle through
equidistant from
.
is a point on the equator. Show that the great circle through
and
bisects the angle
in the spherical triangle
(a spherical triangle has great circle arcs as sides).
Hint
Draw a large diagram. A nice, large, and precise diagram. Note that drawing a sphere entails drawing a circle and then a dashed circle (preferably of a different color) perpendicular (in the plane) to the original circle.
Solution
Since is the north pole, we define the Earth with a sphere of radius one in space with
and sphere center
We then pick point
on the sphere and define the
as the plane that contains great circle points
,
, and
with the
perpendicular to the
and in the direction of
.
Using this coordinate system and ,
, and
axes
where
is the angle from the
to
or latitude on this sphere with
Since and
are points on a great circle through
equidistant from
, then
Since is a point on the equator, then
where
is the angle on the
from the origin to
or longitude on this sphere with
We note that vectors from the origin to points ,
,
, and
are all unit vectors because all those points are on the unit sphere.
So, we're going to define points ,
,
, and
as unit vectors with their coordinates.
We also define the following vectors as follows:
Vector is the unit vector in the direction of arc
and tangent to the great circle of
at
~Tomas Diaz
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1979 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.