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Math Challenge III Geometry 2.4

 
 
ShiJames的头像
Math Challenge III Geometry 2.4
ShiJames - 2020年10月2日 Friday 11:33
 
For this problem, I think that I have to rotate something and then create a triangle and apply triangle inequality, but I am not sure which one to rotate. Does anyone have any hints? Thank you.
 
WangDr. Kevin的头像
Re: Math Challenge III Geometry 2.4
WangDr. Kevin - 2020年10月2日 Friday 14:22
 

Here is the question:

In convex quadrilateral $ABCD$, let $M$ be the midpoint of $\overline{BC}$, and $\angle AMD=120^\circ$.  Show that $AB + \dfrac{1}{2}BC + CD \geq AD$.

When rotation works, usually there are some equal lengths, such as the sides of a square of an equilateral triangle.  In this case, the only equal lengths are $BM$ and $MC$, and it is not easy to see how a rotation from $BM$ to $MC$ helps with the inequality.  Can you consider reflection?  Also, it might not be triangle inequality because there are three terms on the left hand side; of course the principle that "the straight line is the shortest path between two given points" is still at play.