2018 AMC 12B Problems/Problem 21
Contents
[hide]Problem
In with side lengths
,
, and
, let
and
denote the circumcenter and incenter, respectively. A circle with center
is tangent to the legs
and
and to the circumcircle of
. What is the area of
?
Diagram
~MRENTHUSIASM
Solution 1 (Coordinate Geometry)
In this solution, let the brackets denote areas.
We place the diagram in the coordinate plane: Let and
Since is a right triangle with
its circumcenter is the midpoint of
from which
Note that the circumradius of
is
Let denote the semiperimeter of
The inradius of
is
from which
Since is also tangent to both coordinate axes, its center is at
and its radius is
for some positive number
Let
be the point of tangency of
and
As
and
are both perpendicular to the common tangent line at
we conclude that
and
are collinear. It follows that
or
Solving this equation, we have
from which
Finally, we apply the Shoelace Theorem to
Remark
Alternatively, we can use as the base and the distance from
to
as the height for
- By the Distance Formula, we have
- The equation of
is
so the distance from
to
is
Therefore, we get
~pieater314159 ~MRENTHUSIASM
Solution 2
Let points Q and R be the points of tangency between the incircle and lines and
. Notice that
is half of
. Let
= θ. Using the half angle tangent formula and remembering that cos(θ) =
, we find that tan(θ/2) = sqrt((1-cosθ)/(1+cosθ)) =
=
. Now we can find the length of BC in terms of the radius of the incircle, which will be r. Using the knowledge that
is right with some trigonometry, we find that
is 3r/2. We also find that
is the r, and so we can create the equation
+
=
-> 3r/2 + r = 5 -> 5r/2. We conclude r = 2. We now accept the fact that quadrilateral
is a kite, so
=
= 3. We also know that O lies on
and divides
in half. Next we determine that
=
-
= 13/2 - 3 = 7/2. We also know that
has a measure of 2 since it's the radius of circle I
We know very little about the placement of , so we work on that. First, we can conclude that
is part of
. We guess that
has to be higher up than
. We find the area of
to be 7/2, which is the highest answer choice. I don't think that
is below
, but if it were to go past
then the answer would be GREATER than 7/2. This means
is on
. Therefore
. The answer must be
.
~me
~Judokid (revisions)
Solution 3
The circle with center is the
-mixtilinear incircle
of
. Let
be the intersection between
and the circumcircle
. Then, there is a homothety centered at
sending
to
. As such, the tangent
gets sent to a parallel tangent to
, which thus must be the arc midpoint
of arc
. Thus, by inscribed angle theorem
and
intersect at
, so
passes through
where
,
are the tangency points of
with
and
by Pascal's Theorem. Thus, we see since
,
. Set up a coordinate plane and apply Shoelace to obtain
.
See Also
2018 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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