The three figures

Regarding dot grids, a recurring theme that came up again last week, Salva Fuster says: “I would say that for an nxn grid we can join all the dots without lifting the pencil from the paper or going over the same line twice with 2(n-1) straight lines, except in the case of the 2x2 grid, which would require three lines. However, it would have to be demonstrated that it is impossible to do it with fewer lines.”

By the way, for the 4x4 grid, there's a different solution than the one shown in the figure, with several vertices of the broken line outside the grid. Can you find it? (Hint: it's an elegant symmetrical solution.)

And with respect to the rectangle and the moving pieces, Rafael Granero says: “C moves to the middle of its long side (parallel to AB).

B moves parallel to the new line AC until it is level with the side where C is, thus aligning itself with C. A moves parallel to line BC until it is halfway along its long side. And now we have points A and C halfway along the long sides, and B along the hills of Úbeda.”

And B will remain there no matter what we do, since it is impossible to place the three pieces at the midpoints of three of the sides of the rectangle. When one of the vertices moves parallel to the line determined by the other two, the area of ​​triangle ABC remains constant, since neither the base nor the length of the height vary, and its value is 1/2 of the area of ​​the rectangle, while the area of ​​a triangle with its vertices at the midpoints of three sides would be 1/4 of that of the rectangle.

Regarding the radii of exinscribed circles, Luis Ortiz proposes an ingenious approach: “The radii of the exinscribed circles of a triangle with sides 3, 4 and 5 have lengths of 2, 3 and 6, a problem that will be solved below by equations. The sides of the triangle satisfy the Pythagorean theorem, so it is a right triangle. The legs are made to coincide with the x and y axes of a Cartesian system, such that the vertices of the triangle have the following coordinates: A(0,0), B(4,0) and C(0,3)”. From here on, the proof is simple (I do not include it in full for space reasons).

Circles, triangles and rectangles everywhere

In recent weeks we've talked about circles, triangles, and rectangles, which, coincidentally (or maybe not), are the three geometric figures we see everywhere.

The omnipresence of wheels on all types of vehicles in our motorized society would be enough to explain the predominance of circles; but we also find them on many other objects, such as jar and manhole covers or saber blades.

It's obvious that vehicle wheels and the lids of many jars must be round, since they must rotate in a self-aligned manner; but manhole covers could be square, or rectangular, or elliptical... However, there are at least three compelling reasons (pun intended) for them to be circular. What are these three reasons? Or are there more than three?

And without needing to be an expert in bladed weapons, we can safely say that the curvature of a saber or katana blade is an arc of a circle. Why?

As for triangles, there's a compelling reason (pun intended) why we see them constantly in all kinds of structures, from power line towers to geodesic domes, including the Eiffel Tower itself. What's that compelling reason?

And it's no coincidence that boxes, bricks, walls, sheets of paper, and so many other things are rectangular (or orthohedral, which is the same thing in 3D), to the point that Le Corbusier said that the right angle is our pact of solidarity with nature. What did he mean?