Question 1120393
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No; it is not possible.<br>
Consider first the fact that an angle bisector in a triangle divides the opposite side into two parts in the same ratio as the lengths of the two sides of the angle.<br>
In triangle ROE, OD is the angle bisector.  If RD=1 and DE=2, OE is twice the length of OR.  Let OR=x and OE=2x.<br>
In triangle DOS, OE is the angle bisector.  If DE=2 and ES=4, OS is twice the length of OD.  Let OD=y and OS=2y.<br>
Now use Stewart's Theorem in triangles ROE and DOS to get two equations relating x and y.<br>
Triangle ROE:
{{{2x^2+4x^2 = 3y^2+6}}}
{{{3y^2 = 6x^2-6}}}
{{{y^2 = 2x^2-2}}}<br>
Triangle DOS:
{{{4y^2+8y^2 = 24x^2+48}}}
{{{12y^2 = 24x^2+48}}}
{{{y^2 = 2x^2+4}}}<br>
But now we have two equations, based on the given information, that say in one case that y^2=2x^2-2 and in the other case y^2 = 2x^2+4.<br>
So the given conditions lead us to two equations that are incompatible; that means the given conditions are not possible.