PCA + PBC + PBA = ABC
Therefore, 0.5 x side x sum of lengths of perpendiculars from P = 0.5 x side x altitude
This was the first proof that came to mind. Is there a shorter, sweeter one?
Sroyon | 25 Jul 2009 - 17:24 PDT
That's the shortest/sweetest one I am aware of.
Tommy | 25 Jul 2009 - 17:42 PDT
Ok,I admit it. I spent a fair amount of time on this and couldn't find the answer. I doubt there's a shorter answer than Sryon's unless you accept the one my roomate proposed -"obviously they are equal,just look at them".
Rahul | 28 Jul 2009 - 11:36 PDT
maths is hard!
shadows | 29 Jul 2009 - 15:29 PDT
If anyone wondered how to do it with analytical geometry:
Putting point A in the drawing at the origin and point P at (x,y), the distance from P to AB is -sqrt(0.75)*x-y/2 and the distance from P to AC is sqrt(0.75)*x-y/2. (Derived from dot products of <x,y> with unit vectors perpendicular to the lines.) The rest is easy. :)
Tommy | 29 Jul 2009 - 16:19 PDT