Given any triangle $\triangle ABC$, and given one of its side, we can draw two lines perpendicular to that side passing through its two vertices. If we do this construction for each side, we obtain the points $D,E,F$ where two of these perpendicular lines meet at the minimum distance to each side.

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These three points can be used to build three triangles on each side of the starting triangle.

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The conjecture is that

The sum of the areas of the triangles $\triangle AFB$, $\triangle BDC$, and $\triangle CEA$ is equal to the area of $\triangle ABC$.

This is likely an obvious and very well known result. But I cannot find an easy proof of this. Therefore I apologize for possible triviality, and I thank you for any suggestion.

  • 1
    I am not too sure but did you consider that there might be a point inside the triangle, $G$ such that $AFBG$, $BDCG$ and $AECG$ are parallelograms? – Raptor Nov 9 at 7:58
  • I agree with Raptor, and would guess it is the center of the inner circle. i.e. intersection of angle symmetrals. however, have you also tried it for triangles with an obtuse angle? there should be weird stuff happening there, since one of you orthogonals actually will go into the triangle! – Enkidu Nov 9 at 8:14
  • The heights are meeting at that point – Moti Nov 9 at 8:14
  • Thanks all for your comments! – Andrea Prunotto Nov 9 at 9:38
  • @Raptor@Enkidu. Working on it! – Andrea Prunotto Nov 9 at 9:52
up vote 29 down vote accepted

Draw the orthocenter. You get three parallelograms which immediately provide the answer.

  • Nice and easy solution! – YiFan Nov 9 at 8:31

The answer of Moti is perfect. Note, though, that this also means that this property holds only if the orthocenter is inside the triangle, otherwise the external triangles overlap and the required property does not hold any more.

Internal orthocenter.

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External orthocenter.

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  • 2
    No, I do not think so. Maybe that we must consider the oriented area of the triangles build with respect to each side. I guess that the sum of the oriented areas of these triangles (some positive some negative) still equals the area of the original triangle. – Francesco Iovine Nov 9 at 11:51
  • 2
    Signed areas do, indeed, save the result. – Blue Nov 9 at 11:54
  • @FrancescoIovine That the sum of the oriented areas is equal to the original triangle's area is also easily shown using the orthocenter. – Vaelus Nov 9 at 12:57

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