2003 AIME II Problems/Problem 7
Find the area of rhombus given that the circumradii of triangles and are and , respectively.
The diagonals of the rhombus perpendicularly bisect each other. Call half of diagonal BD and half of diagonal AC . The length of the four sides of the rhombus is .
The area of any triangle can be expressed as , where , , and are the sides and is the circumradius. Thus, the area of is . Also, the area of is . Setting these two expressions equal to each other and simplifying gives . Substitution yields and , so the area of the rhombus is .
Let . Let . By the extended law of sines, Since , , so Hence . Solving , . Thus The height of the rhombus is , so we want
Video Solution by Sal Khan
https://www.youtube.com/watch?v=jpKjXtywTlQ&list=PLSQl0a2vh4HCtW1EiNlfW_YoNAA38D0l4&index=16 - AMBRIGGS
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