I. Model "12345"

tan(a+β)=(tanα+tanβ)/( 1-tanαtanβ)

tan(a-β)=(tanα-tanβ)/( 1+tanαtanβ)

β = 45+A

Narrow sense:

If tanα= 1/2, then tan (45+a) = tan β = 3.

If tanα= 1/3, then tan (45+a) = tan β = 2.

Generalization:

If tanα=b/a, then tan β=(a-b)/(a+b)

A+β = 45。

Narrow sense:

If tanα= 1/2, then tan β= 1/3.

Generalization:

If tanα=b/a, then tan β=(a-b)/(a+b)

Half-angle characteristics of "345" triangle

If tan2alpha = 3/4, then tan2α=3/4 kloc-0//3.

If tan2alpha = 4/3, then tan2α=4/3 kloc-0//2.

Second, the narrow and broad "12345" model

The "12345" theorem in two cases is derived from the formula of two angles.

1 2345 theorem when α+β = 45.

What if? α+β=45 :

① tan(α)=2, let tan(β)=t,

Then, tan (45) = tan (α+β) =1= (tan (α)+tan (β))/(1-tan (α) tan (β)) = (1/2+t)/

→ 1-t/2 = 1/2+t→t = 1/3

② tan(α)=3, let tan(β)=t,

Then, tan (45) = tan (α+β) =1= (tan (α)+tan (β))/(1-tan (α) tan (β)) = (1/3+t)/

→ 1-t/3 = 1/3+t→t = 1/2

2. Theorem 12345 is in α+45.

If α+45 =β:

①tan(α)= 1/2, let tan(β)=t,

Then, Tan (β) = Tan (α+45) = (1/2+1)/(1-kloc-0//2) = 3.

②tan(α)= 1/3, let tan(β)=t,

Then, Tan (β) = Tan (α+45) = (1/3+1)/(1-kloc-0//3) = 2.

Summary:

③ tan(α)=b/a, let tan(β)=t,

Then, Tan (β) = Tan (α+45) = (b/a+1)/(1-b/a) = ((b-a)/(a-b)/a) = (a+b).

* Note: Due to A-B; At 45 degrees, the tan value is greater than 1, that is, the denominator is less than the numerator, so the generalized denominator is a-b.

Application of 12345 model

12345 model is a model composed of similar triangles and five similar triangles, and the vertices of each triangle are 1, 2, 3, 4 and 5 respectively.

Every triangle in this model is similar, and their side length ratio is 1:2:3:4:5. In other words, if we know the side length of any triangle, we can calculate the side lengths of the other four triangles.

This model is widely used in fields such as architecture, engineering and geography. , it can be used to calculate the distance, height, angle and other issues. By measuring the side length of a triangle in the model, the side lengths of other triangles can be calculated quickly, thus solving practical problems.

In addition to its application in practical problems, 12345 model has some interesting mathematical properties. For example, the height of each triangle in the model is the sum of the side lengths of other triangles, which can be proved by similar triangles's properties and triangle area formula.

12345 model can also be used to solve some interesting geometric problems. For example, we can draw a line between any two vertices in the model, so that we can get some interesting geometric figures, such as pentagons, quadrangles and so on. These figures can also calculate their area, perimeter, etc. Through the nature of similar triangles.