Difference between revisions of "2021 Fall AMC 12B Problems/Problem 24"
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By the Inscribed Angle Theorem and the definition of angle bisectors note that<cmath>\angle ABD=\angle ABC=\angle AEC\ \text{and}\ \angle BAD=\angle DAC=\angle EAC</cmath>so <math>\triangle ABD\sim\triangle AEC</math>. Therefore <math>\frac{AB}{AD}=\frac{AE}{AC}\rightarrow AB\cdot AC=AD\cdot AE</math>. By PoP, we can also express <math>AD\cdot AE</math> as <math>AB\cdot AF,</math> so <math>AB\cdot AC=AB\cdot AF\rightarrow AC=AF=20</math> and <math>BF=20-AB=20-11=9</math>. Let <math>CF=x</math>. Applying Stewart’s theorem on <math>\triangle ACF</math> with cevian <math>\overrightarrow{CB},</math> we have | By the Inscribed Angle Theorem and the definition of angle bisectors note that<cmath>\angle ABD=\angle ABC=\angle AEC\ \text{and}\ \angle BAD=\angle DAC=\angle EAC</cmath>so <math>\triangle ABD\sim\triangle AEC</math>. Therefore <math>\frac{AB}{AD}=\frac{AE}{AC}\rightarrow AB\cdot AC=AD\cdot AE</math>. By PoP, we can also express <math>AD\cdot AE</math> as <math>AB\cdot AF,</math> so <math>AB\cdot AC=AB\cdot AF\rightarrow AC=AF=20</math> and <math>BF=20-AB=20-11=9</math>. Let <math>CF=x</math>. Applying Stewart’s theorem on <math>\triangle ACF</math> with cevian <math>\overrightarrow{CB},</math> we have | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
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x&=\boxed{\textbf{(C)} ~30}. | x&=\boxed{\textbf{(C)} ~30}. | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | |||
~Punxsutawney Phil | ~Punxsutawney Phil | ||
Revision as of 01:17, 28 November 2021
Problem
Triangle has side lengths
, and
. The bisector of
intersects
in point
, and intersects the circumcircle of
in point
. The circumcircle of
intersects the line
in points
and
. What is
?
Solution 1
Claim:
Proof: Note that and
meaning that our claim is true by AA similarity.
Because of this similarity, we have that by Power of a Point. Thus,
Now, note that and plug into Law of Cosines to find the angle's cosine:
So, we observe that we can use Law of Cosines again to find :
- kevinmathz
Solution 2
By the Inscribed Angle Theorem and the definition of angle bisectors note thatso
. Therefore
. By PoP, we can also express
as
so
and
. Let
. Applying Stewart’s theorem on
with cevian
we have
~Punxsutawney Phil
Solution 3
This solution is based on this figure: Image:2021_AMC_12B_(Nov)_Problem_24,_sol.png
Denote by the circumcenter of
.
Denote by
the circumradius of
.
In , following from the law of cosines, we have
For , we have
The fourth equality follows from the property that
,
,
are concyclic.
The fifth and the ninth equalities follow from the property that
,
,
,
are concyclic.
Because bisects
, following from the angle bisector theorem, we have
Hence, .
In , following from the law of cosines, we have
and
Hence, and
.
Hence,
.
Now, we are ready to compute whose expression is given in Equation (2).
We get
.
Now, we can compute whose expression is given in Equation (1).
We have
.
Therefore, the answer is .
~Steven Chen (www.professorchenedu.com)
See Also
2021 Fall AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.