Problem 21
Let ABC be any triangle. Points D and E are taken on AB and BC respectively such that BD/DA = CE/BE. Prove that:
ar(BDE)/ar(ABC) ≤ 1/4
Here’s a simulation that demonstrates what needs to be proven.
Try dragging the point D and the vertices of the triangle. Is the maximum value of ar(BDE)/ar(ABC) equal to 1/4?
Solution
Here’s what you’ll need to know to be able to solve the problem.
Area of a Triangle
The area of a triangle in terms of its adjacent side lengths and included angle is given by:
Area = 1/2 x a x b x sinC
where a and b are the lengths of the adjacent sides and C is the included angle
Arithmetic-Geometric Mean Inequality
For any n positive numbers a1, a2, a3, … , an, we have:
(a1 + a2 + a3 + … + an)/n ≥ (a1.a2.a3…an)1/n
That is, their arithmetic mean is always greater than or equal to their geometric mean.
Now, to proceed, try finding the areas of the two triangles using the above formula. Then, divide them to find the ratio and try applying the above inequality to find the maximum value.
Were you able to solve it using the above hints? Or, did you solve using any other method? Or, do you need some more help? You can reach out at the Telegram group.
Problem Source: CutTheKnot
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