Proof without words of a simple conjecture about any triangle

Given the midpoint (or centroid) $$D$$ of any triangle $$\triangle ABC$$, we build three squares on the three segments connecting $$D$$ with the three vertices. Then, we consider the centers $$K,L,M$$ of the three squares. My conjecture is that

The area of the triangle $$\triangle KLM$$ is equal to half of the area of the triangle $$\triangle ABC$$. This is for sure a well known result (well, if true!). In this case, sorry for the trivial problem!

However, It would be great to have suggestions for developing a proof without words of such simple claim (again, if true), i.e. avoiding trigonometry, etc. Thanks for your help!

EDIT: The conjecture can be easily extended to any regular polygon built on the described segments (e.g. equilateral triangles yield to $$1/3$$ of the $$\triangle ABC$$ area, etc.).

EDIT (2): The (extended) conjecture appears to be true also by building the segments starting from the orthocenter (red, left), instead of the centroid (grey, right). The area of the final triangle $$\triangle KLM$$ is however the same! • Would you know a reference for the proof of this please? Thanks. Nov 11 '18 at 17:36
• @NoChance Well, I am not fully sure that this claim is true. Therefore I have no clue of a proof. Maybe should I remove the tag "alternative-proof"? Sorry for confusion.
– user559615
Nov 11 '18 at 17:38
• Thank you for your response, somehow I though there is a proof because of the wording "This is for sure a well known result". Very interesting idea. Nov 11 '18 at 17:40
• A proof without words generally takes a known proof, and finds a geometric representation of what's happening. If you don't yet know that it's true, then finding a proof would be a better place to start than looking for a specific type of proof. Nov 11 '18 at 20:44
• @Teepeemm Sure, I definitely agree. However, now greedoid provided a very nice and simple proof. So we can maybe focus on the proof without words! ; )
– user559615
Nov 11 '18 at 21:25

Observe a spiral similarity with center at $$D$$ which takes $$A$$ to $$K$$. Then it takes $$B$$ to $$M$$ and $$C$$ to $$L$$. So this map takes triangle $$ABC$$ to triangle $$KML$$ which means that they are similary with dilatation coefficient $$k={\sqrt{2}\over 2}$$ So the ratio of the areas is $$k^2 =1/2$$.

• Thanks, greedoid for your always neat answers. I thought there could be some even easier solution, tangram/mosaic-like. But I am not sure. Thanks again however!
– user559615
Nov 11 '18 at 17:45
• Oh, sorry, I'm a bad reader (and writter). Anyway you should develop whole theory before you can play with ,,proofs without words''. You can not do anything just by the picture.
– Aqua
Nov 11 '18 at 17:50
• True. I see your point. Say, I thought it should have been not too difficult to prove this (e.g. with trigonometry), but I was looking for a more visual approach. But, true: first one has to have a proof! Therefore, thanks again for yours!
– user559615
Nov 11 '18 at 18:26
• I've just realized that the conjecture can be extended by building any regular polygon on these segments (with equilateral triangles, the centers define a triangle of area $1/3$ of the initial triangle, with hexagons, the areas coincide and so on). I think your proof can be nicely generalized!
– user559615
Nov 11 '18 at 21:51

The two triangles share a centroid. The small triangle vertices are one half of each square's diagonal while the larger triangle has the length of a side. The triangles are similar since lengths are proportional by a factor of sqrt(2) which we square for an area comparison of 2:1 in favor of the larger one.

• Perhaps as a visual you can show the construction of the squares, show a copy of each side as it contracts in length and rotates into position along the diagonal to the shared centroid. Then connect the vertices to show the small triangle. I believe this would have the desired effect except for deriving the factor of sqrt(2) which would require a visual proof of this case of the pythagorean theorem. I imagine there are several of those available for such a popular theorem. Nov 11 '18 at 23:16
• Thanks Nathan. And welcome on MSE!
– user559615
Nov 12 '18 at 4:31
• Thanks for the interesting challenge and welcome. It just popped up on my Google feed! Nov 13 '18 at 18:46

This could be an idea for a "proof without words": In the special case where $$\triangle ABC$$ is right and isosceles, it seems true almost visually that, in triangles $$KLM$$ and $$ABC$$, base$$KM=BC$$and altitude$$BA=2BE$$making$$\triangle ABC=2\triangle KLM$$ On the other hand, if we build the squares on the sides of $$\triangle ABC$$, as in the figure below, the reverse appears to happen:$$\triangle KLM=2\triangle ABC$$since base$$KL=AC$$and altitude$$BM=2BE$$ OP's conjecture seems, as some are suggesting, a particular result of a larger general theory?

The above figures suggest perhaps the further question, whether a "proof without words" is ever really possible; it seems words are implicitly present in any train of thought, and a proposition that was truly self-evident would not be a proof strictly speaking but rather a basis or element of a proof.