Hi @sergeyk this looks really very interesting and I just got time yesterday to play in the playground of Geogebra.
I tried to do first task but I was unable to do similar as your output appears.
I hided the perpendiculars by adding segments inside the triangle but when I extend the corners the dotted lines does not appear as yours but nothing appears.
Can you please guide me how to do it? Indeed in the new tasks there is no similar thing but it looks very interesting to learn it. If you have then please guide me.
I think it would be better to publish your work even if not all the tasks are completed.
That way I would know what exactly you are doing and what is not.
Try to complete some tasks from past lessons and write a post.
As for this construction, the winners described it very well in their works. And specifically this triangle is present in both the first and second homework tasks.
The main idea is to build additional segments, straight lines and hide unnecessary ones. After the heights are built, you should make the triangle obtuse and connect the base of the height with the vertex with an additional segment, and so on with each height
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