1961 AHSME Problems/Problem 11
Problem
Two tangents are drawn to a circle from an exterior point 
; they touch the circle at points 
and 
respectively.
A third tangent intersects segment 
in 
and 
in 
, and touches the circle at 
. If 
, then the perimeter of 
is
Solution
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Draw the diagram as shown. Note that the two tangent lines from a single outside point of a circle have the exact same length, so 
, 
, and 
.
The perimeter of the triangle is 
. Note that 
, so from substitution, the perimeter is
![\[AP + PQ + QR + AR\]](http://latex.artofproblemsolving.com/5/f/d/5fd2b44a5bca7a5f2a1b11ce33eba2a6b30ec050.png)
![\[AP + BP + CR + AR\]](http://latex.artofproblemsolving.com/e/8/3/e83a9d7d135210ae7da5bd0eff3b3c38ca787750.png)
![\[AB + AC\]](http://latex.artofproblemsolving.com/2/3/f/23f0fec0219ebb0a5460ef26d740c0425a4e3a31.png)
Thus, the perimeter of the triangle is 
, so the answer is 
.
Solution 2 (Non-rigorous)
Since can be anywhere on the circle between and , it can basically be "on top" of . Then will be at the same point as , so 
form a degenerate triable with side length . So its perimeter will be . Since 
and 
by power of a point, as 
and 
decrease in length, 
will "grow" to compensate, so the perimeter will stay constant with a value of 
.
We can also skip verifying that the perimeter will stay constant, since it seems unlikely that MAA would create a question with 
as the answer.
~jd9
See Also
| 1961 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
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| All AHSME Problems and Solutions | ||
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