ogeometrie: decembrie 2023

marți, 12 decembrie 2023

PROBLEM MA241 CRUX MATHEMATICORUM V49 No 9

Pag. 462 - En

Pag. 464 - Fr

ANSWER CiP

$$n=33$$

Solution CiP

The same hexagon can be inscribed in the following equilateral triangle with side length 33.

There is no other option. Indeed, a certain segment of the hexagon, let's say the one of length 4, can only be placed in two places: on the basis of the equilateral triangle or inside an angle.

The base and the angle can be any of the three, but due to the symmetries of the equilateral triangle, only one or the other of the two inscriptions is obtained.

$\blacksquare$

luni, 11 decembrie 2023

The world's simplest demonstration of the Pythagorean Theorem that, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs

The world's simplest

demonstration of the Pythagorean Theorem that,

in a right triangle,

the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.

So if we build a square on the two legs and another on the hypotenuse, then the sum of the areas of the first two is equal to the third. The figure is classic.

Instead of squares, we can build any three similar figures on the three sides. For example semicircles, as in the first figure.

AHA ! Then we can take the height $AD$ of the triangle and automatically similar triangles are formed on each of the three sides : $\Delta ABC$ on the side $[BC]$, $\Delta ADC$ on the side $[AC]$, $\Delta ADB$ on the side $[AB]$. These triangles are similar, having a right angle and a common acute angle. Moreover, there is an obvious relationship between the areas, which is marked on the figure. Or this is precisely the Pythagorean Theorem.