Let ABCDEF be a convex hexagon in which the diagonals AD, BE, CF are concurrent at O.

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Asked: 2023-11-09T03:33:16+05:30 In: Other
Let ABCDEF be a convex hexagon in which the diagonals AD, BE, CF are concurrent at O.

Hi, I have a question and I hope anyone could answer it:

Let ABCDEF be a convex hexagon in which the diagonals AD, BE, CF are concurrent at O. Suppose the area of triangle OAF is the geometric mean of those of OAB and OEF and the area of triangle OBC is the geometric mean of those of OAB and OCD. Prove that the area of triangle OED is the geometric mean of those of OCD and OEF.

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    2. Answer:

      Let OA=a,OB=b,OC=c,OD=d,OE=e,OF=f,[OAB]=x,[OCD]=y,[OEF]=z,[ODE]=u,[OFA]=v

      and [OBC]=w. We are given that v2=zx,w2=xy and we have to prove that u2=yz. 
      Since ∠AOB=∠DOE, we have
      yv​=21​absin∠AOB21​desin∠DOE​=abde​
      Similarly, we obtain    yv​=cdfa​,zw​=efbc​
      Multiplying, these three equalities, we get uvw=xyz. Hence
      x2y2z2=u2v2w2=u2(zx)(xy).
      This gives u2=yz, as desired.
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