id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
american_scientific_journal-757	Nigo, Jemal Kemal	An Alternate Method to Find Area of Triangle from its Vertices alone in a Plane	2015	12	.pdf	application/pdf	3285	149	67	(2015) Volume 13, No 1, pp 49-60 ( )( ) ( )( ) ( )9 9 332313322212312111 321321 yxyxyxyxyxyxyxyxyx yxyyyxxxyx iicc ++++++++= =++++= ∑∑ It has been stated in [1] that the area of the triangle is given by [ ] ( )10 2 1 2 1 231231211332 1312 1312 yxyxyxyxyxyx yyyy xxxx A −−−++= −− −− ±= Thus rewriting the area using ( )9 and ( )10 we have [ ]jiiiii yxyxyx yxyxyxyx yxyxyxyxyxyxyxyxyxyxyx A ∑∑∑∑ −−=       −−−− −−++++++++ = 2 2 1 2222 1 23123133 2211332313322212312111 Where ( )( )321321 yyyxxxyx ii ++++=∑∑ , 332211 yxyxyxyx ii ++=∑ and 122331 3 2cos2 31 3 2cos2 31 yxyxyxxyyx jji j ij jji j ji ++== ∑∑      −= ≤≤      −= ≤≤ ππ Theorem 2: Theorems Theorem 1: The area A of a triangle having vertices ( ) ( )2211 ,,, yxQyxP and ( ),, 33 yxR is given by       −      −            = ∑∑∑∑ ==== j i ii i i i i i i yxyxyxA 3 1 3 1 3 1 3 1 2 2 1 Or for short [ ]jiiiii yxyxyxA ∑∑∑∑ −−= 2 2 1 Where ( )( )321321 yyyxxxyx ii ++++=∑∑ , 332211 yxyxyxyx ii ++=∑ and 122331 yxyxyxyx ji ++=∑ Proof:	cache/american_scientific_journal-757.pdf	txt/american_scientific_journal-757.txt