Challenging Geometry Puzzle: Problem 1595. Share your solution by posting it in the comment box provided.
Audience: Mathematics Education - K-12 Schools, Honors Geometry, and College Level.
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* Geometry Problem 1595 *
ReplyDeleteWe set
AE=x, ED=y, BC=x+y, AB=2*A2/x, DF=2*A3/y, FC=2*A1/(x+y).
From the relationship 2*A2/x = 2*A3/y + 2*A1/(x+y) it follows that:
y = (x/2)( SQR{ [ (A2-A3-A1)^2/A2^2 ] + 4*A3/A2 } - (A2-A3-A1)/A2 )
The lengths of all sides of the rectangle now depend only on an unknown factor x, which is eliminated when we multiply the sides AB*BC = (2*A2/x)(x+y) to get the area of the rectangle. After simplification, we get:
Area ABCD = A1+A2+A3 + SQR{(A2-A3-A1)^2 + 4*A2*A3}
I checked this relationship with Geogebra and it seems correct. I guess the requested proof must be close.