I was recently presented with this problem:
I solved this problem by first drawing up the parallelogram, like this:
Then I added a few extra lines, like this:
adding the height h and a point P.
After that I set a condition that and stated that and also the fact that
, which I'm too lazy to prove right now, as I'm still struggling with latex.
After this I used the pythagorean theorem to show that which means that
As an effect, this means that , as clearly is bigger than
Then, I just made myself an inequality.
And then finally
, which I was supposed to show.
My question is if I've done everything right, as the book in which I found the problem solves it with trigonometry, and the section for the problem was a section about trigonometry. If my solution is incorrect I need to know why.
It's correct. :)
but there is a geometrical proof i got.
Drop perpendicular from B to AD and call the foot of the perpendicular as K. Let AC meet BD at L.
then |AK|=|KD| and |AL|=|LC|.
Let AL and BK meet at G. G is the centroid since AL and BK are medians.
To prove: |AD|/|AC| < 2/3
that is, |AK|/|AL| < 2/3.
Now |AK|<|AG| since AG is the hypotenuse of triangle-AGK.(AGK is a right angled triangle).
so |AK|/|AL| < |AG|/|AL|.
we know that |AG|/|AL|=2/3 from elementary geometry because G is the centroid. which proves the result.