2021 Fall AMC 10A Problems/Problem 15
Isosceles triangle 
has 
, and a circle with radius 
is tangent to line 
at 
and to line 
at 
. What is the area of the circle that passes through vertices 
, , and 
Solution 1
Let the center of the first circle be 
By Pythagorean Theorem,
![\[AO=\sqrt{(3\sqrt{6})^2+(5\sqrt{2})^2}=2 \sqrt{26}\]](http://latex.artofproblemsolving.com/f/a/a/faab0344e40b6da42c07a30d261447e99f9aa063.png)
Now, notice that since 
is 
degrees, so arc 
is 
degrees and is the diameter. Thus, the radius is 
so the area is 
- kante314
Solution 2 (Similar Triangles)
![[asy] import olympiad; unitsize(50); pair A,B,C,D,E,O; A=origin; B=(2,3); C=(-2,3); D=(6,9); E=(-6,9); O=circumcenter(A,B,C); // olympiad - circumcenter draw(A--B--C--cycle); dot(O); draw(circumcircle(A,B,C)); // olympiad - circumcircle draw(incircle(A,D,E)); label("$O$",O,S); label("$A$",A,S); label("$B$",B,E); label("$C$",C,W); label("$3\sqrt{6}$",(1.25,1),S); label("$3\sqrt{6}$",(-1.25,1),S); [/asy]](http://latex.artofproblemsolving.com/8/a/3/8a36f0546545319b166f935f9a63a4cc9fe17767.png)
Solution in Progress
~KingRavi
See Also
| 2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 14 |
Followed by Problem 16 | |
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| All AMC 10 Problems and Solutions | ||
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