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enter image description here

Given the parallelogram ABCD, $\frac{AD}{AE}$ = $\frac{BF}{FA}$ and $\frac{area of \triangle ADF}{areaof \triangle AEF}$ = $\frac{AD}{AE}$ (or in other words $\frac{area of \triangle ADF}{areaof \triangle AEF}$ = $\frac{BF}{FA}$), prove that area of $\triangle$ADF = area of $\triangle$BEF.

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  • $\begingroup$ Show please your attempts. $\endgroup$ – Michael Rozenberg Jul 8 at 20:29
  • $\begingroup$ Sure. All I could figure out was that the base of $\triangle$BEF and the base of $\triangle$ADF are the proportion $\frac{BF}{FA}$. I also know that the heights of the two triangles in question are in proportion to $\frac{AD}{AE}$, as they are the same heights found in the above proportion. After that, I'm lost. $\endgroup$ – bagrut1 Jul 8 at 20:40
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Let EI the height of the triangle AEF. Also DH the height of the triangle DAF. We know that the ratio between the height of similar triangle is equal to the ratio between the sides. So $\frac{DH}{EI} = \frac{BF}{FA} $. Now we can write $ DH \cdot FA = BF \cdot EI$. $BF \cdot EI$ is the double of the area of the traingle $ EFB$; $DH \cdot FA$ is the double of the area of $ DAF$. So $ DAF = EFB$ (Area).

Do I have to post an image to clarify?

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  • $\begingroup$ Firstly, in your proportion, shouldn't BF and FA be flipped? $\endgroup$ – bagrut1 Jul 8 at 20:57
  • $\begingroup$ Secondly, how did you get that BF* FB is double the area of the triangle? Isn't that just multiplying the base times itself? $\endgroup$ – bagrut1 Jul 8 at 20:58
  • $\begingroup$ Yes, sorry it's EI $\endgroup$ – Matteo Jul 8 at 21:00
  • $\begingroup$ Hi, I still don't understand how you got your initial proportion. Shouldn't the heights of respective triangles be lined up with the heights of respective sides? $\endgroup$ – bagrut1 Jul 8 at 21:02
  • $\begingroup$ DAF and FEA are similar $\endgroup$ – Matteo Jul 8 at 21:04
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I think it is all explained in the figure below. enter image description here

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  • $\begingroup$ What is k in your equations? And I don't understand what you mean when you wrote "AE = k.AD; AF = k.BF"? $\endgroup$ – bagrut1 Jul 9 at 5:33
  • $\begingroup$ k is the ratio AE/AD. k = AE/AD = AF/BF $\endgroup$ – Penguino Jul 9 at 20:46
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You just need to use the obvious theorem:

If two triangles have a vertex in common and the sides opposite to that vertex lie on the same line, then the ratio of their areas is equal to the ratio of the opposite sides.

Hence: $$ {area\triangle BEF\over area\triangle AEF}={BF\over FA}={AD\over AE}= {area\triangle ADF\over area\triangle AEF} $$

EDIT.

Notice that $AD/AE=𝐵𝐹/𝐹𝐴$ is the same as $BC/AE=𝐵𝐹/𝐹𝐴$, and this follows from the similarity of triangles $AEF$ and $CBF$.

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