Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1992 based on Largest Area of Triangle.
Triangle ABC has AB=9 and BC:AC=40:41, find the largest area that this triangle can have.
Ratio
Area
Triangle
Answer: is 820.
AIME I, 1992, Question 13
Coordinate Geometry by Loney
Let the three sides be 9, 40x, 41x
area = \(\frac{1}{4}\sqrt{(81^2-81x^2)(81x^2-1)} \leq \frac{1}{4}\frac{81^2-1}{2}\)
or, \(\frac{1}{4}\frac{81^2-1}{2}=\frac{1}{8}(81-1)(81+1)\)
=(10)(82)
=820.
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