# Beck’s Theorem: Calculate the Area – Without a Difficult Formula

Thus it turns out that the formation of two polygons that follow Beck’s theorem also satisfies the theorem automatically. This can be demonstrated in a similar way: if you join two polygons s1 And the s2 to greater swhere this and one of the smallest (s1) satisfies Beck’s theorem, then does the rest s2. Knowing this, one can now prove that the theorem is valid for all triangles that fulfill the requirements.

### Step 3: The triangles satisfy Beck’s theorem

To do this, you start with a right-angled triangle whose legs are aligned horizontally and vertically. Now you can invert this in its chord and get a rectangle that satisfies Pick’s theorem as already shown: A = I + B/ 2 – 1. Once again one can use that I = 2I + J – 2 where I’ are the interior points of triangles. In addition to: B = 2 (with me + 1) where B’ are the edge points of the triangles. Entering it into the zone one gets: a = 2I + J – 2+ with me + 1 – 1 = 2 (I’ + B’/ 2-1). there a Twice the area of ​​the triangle a’ Corresponding, one can see that both triangles are subject to Beck’s theorem.

Now one just has to show that the other triangles also follow the theorem. It’s easy to do: to do this, you draw the smallest rectangle around a given triangle. Since all right triangles follow Beck’s theorem and all rectangles as well, one can use the rule of composition to show that the triangle under study is also subject to the theorem (since it applies to all other shapes within the rectangle).

See also  The millipede species is named after the American singer Taylor Swift | free press