2001 Pan African MO Problems/Problem 6
Let be a semicircle with centre
and diameter
.A circle
with centre
is drawn, tangent to
, and tangent to
at
. A semicircle
is drawn, with centre
on
, tangent to
and to
. A circle
with centre
is drawn, internally tangent to
and externally tangent to
and
. Prove that
is a rectangle.
Solution
We use Cartesian coordinates, setting , and assume without loss of generality that
is closer to
than to
and that the
-coordinate of
is positive. Then we compute
(because that is the only choice that allows for tangency to both
and the line
). Letting
be the radius of
, we find that
. However, since
and
are tangent, we know that
; using the Pythagorean theorem, we then solve
, which solves to
, so that
.
Finally, we know that must now be uniquely determined. If
was a rectangle, then
would have to be located at
, so we only need check that a circle centered there is tangent to
. Letting
denote the radius of
, we know from tangencies that we must have
. These equations are all satisfied for our desired choice of
and the value
, so we conclude that
is rectangular and we are done.
See also
2001 Pan African MO (Problems) | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Last Problem |
All Pan African MO Problems and Solutions |