[Boutique] Proof of Helen's formula

Article 1: Proof of Helen's formula

In △ABC, ∠A, ∠B, ∠C correspond to sides a, b, c

O is the inner center of the inscribed circle, r is the radius of the inscribed circle, and p is its half circumference.

There are tanA/2tanB/2+tanB/2tanC/2+tanC/2tanA/2=1

r=r

∵r=tanA/2=tanB/2=tanC/2

∴ r

=[++]tanA/2tanB/2tanC/2

=ptanA/2tanB/2tanC/2

=r

∴p^2r^2tanA/2tanB/2tanC/2=pr^3

∴S^2=p^2r^2=/

=p

∴S=√p

Article 2: Proof of Helen's formula

We use the triangular formula and the formula deformation to prove. Let the diagonals of the three sides a, b, and c of the triangle be A, B, and C, respectively, then the cosine theorem is

cosC = /2ab

S=1/2*ab*sinC

=1/2*ab*√

=1/2*ab*√[1-^2/4a^2*b^2]

=1/4*√[4a^2*b^2-^2]

=1/4*√[]

=1/4*√[^2-c^2][c^2-^2]

=1/4*√[]

Let p=/2

Then p=/2, pa=/2, pb=/2, pc=/2,

The above formula = √[/16]

=√[p]

Therefore, the triangle ABC area S = √ [p]

PART 3: Proof of Helen's formula

Qin Jiuyi, a mathematician in the Song Dynasty of China, also proposed the "three oblique accumulation". It is basically the same as Helen's formula. In fact, in the "Nine Chapters of Arithmetic", there is already a triangle formula "half the height of the bottom". When actually measuring the land area, because the area of ​​the land is not a triangle, find it. Not easy. So they thought of the three sides of the triangle. If you do this, it is much more convenient to find the area of ​​the triangle. But how to find the area of ​​the triangle according to the length of the three sides? Until the Southern Song Dynasty, China's famous mathematician Qin Jiuzhen proposed the "three oblique quadrature."

Qin Jiuyu, he called the three sides of the triangle as small oblique, medium oblique and large oblique. "Surgery" is the method. The three-slope quadrature method is to use a small oblique square plus a large oblique square, and send it to the middle oblique square. Take the half of the remainder after the phase subtraction, and take a number by multiplication. The small oblique square is multiplied by the large oblique square, and sent to the above. That one. After the subtraction, the remainder is divided by 4, and the obtained number is taken as "real", and 1 is taken as "隅", and the area is obtained after square rooting.

The so-called "real" and "隅" mean that in the equation px 2 = q, p is "隅" and q is "real". △, a, b, c represent the area of ​​the triangle, large oblique, medium oblique, small oblique, so

q=1/4{a^2*c^2-[/2 ]^2}

When P=1, △ 2=q,

△=√1/4{a^2*c^2-[/2 ]^2}

Factorization

△ ^2=1/16[4a^2c^2-^2]

=1/16[ ^2-b ^2][b^ 2-^ 2]

=1/16

=1/16

=1/16 [2p]

=p

Therefore:

S△=√[p]

Where p=1/2

This is completely consistent with the Helen formula, so this formula is also known as the "Helen-Qin Jiuyi formula."

S=√1/4{a^2*c^2-[/2 ]^2} . where c>b>a.

According to the Helen formula, we can continue to generalize it to the area calculation of the quadrilateral. The following questions:

It is known that the quadrilateral ABCD is a circular inscribed quadrilateral, and AB=BC=4, CD=2, DA=6, and the area of ​​the quadrilateral ABCD is obtained.

Here is the promotion of Helen's formula

S circle inscribed quadrilateral = under the root

Substitute solution s=8√ 3