Geometry Problem

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## Wednesday, February 23, 2011

### Problem 582: Quadrilateral, Diagonals, Incenters, Metric Relations

Labels:
incenter,
metric relations,
quadrilateral,
triangle

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Connect OE,OF,OG,OH

ReplyDeleteWe have E,O,G collinear and F,O,H collinear ( angle bisectors of vertical angles)

We also have EG perpendicular to FH at O ( angle bisectors of 2 supplement angles)

apply Pythagoras therorem we have

EF^2+HG^2=OE^2+OF^2+OH^2+OG^2=EH^2+FG^2

Peter Tran

angle EOF is right angled as the bisectors of a linear pair enclose a right angle. so by applying Pythagoras theorem and associativity property of addition, we have EF^2 +GH^2 = (FO^2 + EO^2) + (HO^2 + GO^2) = (FO^2 + GO^2) + (EO^2 + HO^2) = FG^2 + EH^2.

ReplyDeleteQ. E. D.