perplexus.info :: Numbers : quadrilateral in a hexagon

quadrilateral in a hexagon (Posted on 2024-12-02)

Let ABCDEF be a regular hexagon of unit side length. Line BE is extended to a point P outside of the hexagon such that FP=√3. The line AP intersects the lines BF and EF at points M and N, respectively. Calculate the area of quadrilateral BENM.

No Solution Yet Submitted by Danish Ahmed Khan

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Solution

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Since BF=FP=sqrt(3), triangle BFP is isosceles and angleFBP=angleFPB=30

angleFAE=30

angleAEP=anglePFA=150

AEPF is a parallelogram, so its diagonals bisect and FN=NE=1/2

also EP=1 so BP=3

AFPB is a trapezoid so its diagonals make triangles AMF and PMB similar with size ratio 3:1.

Since FB=sqrt(3), FM=sqrt(3)/4

[BENM] = [BFE] - [FMN]

=1/2*sqrt(3)*1 - 1/2*sqrt(3)/4*1/2

=7sqrt(3)/16

Posted by Jer on 2024-12-04 14:37:20

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