2009 AIME II Problems/Problem 10
Four lighthouses are located at points , , , and . The lighthouse at is kilometers from the lighthouse at , the lighthouse at is kilometers from the lighthouse at , and the lighthouse at is kilometers from the lighthouse at . To an observer at , the angle determined by the lights at and and the angle determined by the lights at and are equal. To an observer at , the angle determined by the lights at and and the angle determined by the lights at and are equal. The number of kilometers from to is given by , where , , and are relatively prime positive integers, and is not divisible by the square of any prime. Find + + .
Video Solution by Punxsutawney Phil
Let be the intersection of and . By the Angle Bisector Theorem, = , so = and = , and + = = , so = , and = . Let be the foot of the altitude from to . It can be seen that triangle is similar to triangle , and triangle is similar to triangle . If = , then = , = , and = . Since + = = , = , and = (by the pythagorean theorem on triangle we sum and ). The answer is + + = .
Extend and to intersect at . Note that since and by ASA congruency we have . Therefore .
By the angle bisector theorem, and . Now we apply Stewart's theorem to find :
and our final answer is .
Notice that by extending and to meet at a point , is isosceles. Now we can do a straightforward coordinate bash. Let , , and , and the equation of line is . Let F be the intersection point of and , and by using the Angle Bisector Theorem: we have . Then the equation of the line through the points and is . Hence the intersection point of and is the point at the coordinates . Using the distance formula, for an answer of .
After drawing a good diagram, we reflect over the line , forming a new point that we'll call . Also, let the intersection of and be point . Point lies on line . Since line bisects , we can use the Angle Bisector Theorem. and , so . Letting the segments be and respectively, we now have . Therefore, . By the Pythagorean Theorem, . Using the Angle Bisector Theorem on , we have that . Substituting in , we have that , so the answer is .
Solution 5 (Angle Bisector Theorem + Law of Cosines)
Let and meet at ; then is an angle bisector of isosceles . Then by the Angle Bisector Theorem, , and . By the Law of Cosines on , we have and the answer is .
Solution 6 (Law of Sines)
Using the law of sines on ,
the answer is .
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