Geometry question

[plain nexus]

Consider the triangle ABC with sides of lengths |AB| = 2, |AC| = 3 and |BC| =
√
7. We take the points D and E on the sides AB and AC, respectively, in such a way that the area of ​​the
triangle ADE is 1/3 of the area of ​​the quadrilateral BCED and the length DE is a minimum. determine
the lengths |AD| and |AE|.

[night valley]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with: ```diff
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[soft night]

This is the question, right?

There are three variables that can change this: a, b and alpha

Or you can call DE, c. And remove the alpha from the variable list

From here you can find the angle at the point A, using a, b, c lengths

$\frac{a^2 + b^2 - c^2}{2ab} = cos(A)$

[unreal bolt]

Oğuzhan

[soft night]

You can use this identity to solve it. But there should be an easier way

[rain acorn]

@plain nexus Note that such line segment is the one that connects the midpoints of AB, AC. That is because in that case DE=BC/2 and height (ADE)=height (ABC)/2. Therefore Area(ADE)=DEheight (ADE)/2=(BCheight (ABC)/2)/4=Area(ABC)/4