Quadrilateral Equals Overlapping Triangles (Posted on 2010-07-23)

Let P and Q be points on the sides AB and CD respectively
of convex quadrilateral ABCD. Prove the following:

    |AP|     |CQ|
If ------ = ------, then [ABCD] = [QAB] + [PCD]
    |PB|     |QD|

where [KLM...XYZ] denotes the area of polygon KLM...XYZ.

	Submitted by Bractals
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Solution:	(Hide)
Join A to C and P to Q and let |AP| |CQ| ------ = ------ = k |PB| |QD| Then, k [QAP] [CAP] [ACQ] [PCQ] ----- = ------- = ------- = ------- = -------. k+1 [QAB] [CAB] [ACD] [PCD] Thus, k+1 [ABCD] = [CAB] + [ACD] = ----- ([CAP] + [ACQ]) k k+1 k+1 = ----- [APCQ] = ----- ([QAP] + [PCQ]) k k = [QAB] + [PCD] ------------------------------------------------------- See Harry's post for an alternate solution.

	Subject	Author	Date
	Solution	Harry	2010-07-26 20:59:54