The main applications of triangle area calculation formula in solving problems are:

Let △ABC, where A, B and C are opposite sides of angles A, B and C respectively, ha is the height of side A, R and R are the radii of the circumscribed circle and inscribed circle of △ABC respectively, and p = (a+b+c), then

S△ABC = aha= ab×sinC = r p

= 2R2sinAsinBsinC =

=

Among them, S△ABC = is the famous Helen formula, which was recorded by the Greek mathematician Helen in Geodesy.

Helen formula has a very important application in solving problems.

First, the deformation of Helen formula

S=

= ①

= ②

= ③

= ④

= ⑤

Second, the proof of Helen formula

Prove Pythagorean Theorem

Analysis: Starting with the most basic calculation formula of triangle S△ABC = aha, Helen's formula is deduced by Pythagorean theorem.

Proof: As shown in figure ha⊥BC, according to Pythagorean theorem, we get:

x = y =

Ha = = =

∴ S△ABC = aha= a× =

At this time, S△ABC is deformation ④, so it is proved.

Certificate 2: Smith Theorem

Analysis: On the basis of the first proof, ha can be obtained directly by using Smith theorem.

Smith theorem: take any point d on the BC side of △ABC,

If BD=u, DC = v and AD = t, then

t 2 =

Proof: from the first proof, u = v =

∴ ha 2 = t 2 = -

∴ S△ABC = aha = a ×

=

This is the deformation ⑤ of S△ABC, so it is proved.

Proof 3: Cosine Theorem

Analysis: According to the deformation 2S =, it is proved by cosine theorem c2 = a2+b2 -2abcosC.

Prove: Prove S =

It is necessary to prove that S =

=

= ab×sinC

At this time, S = ab×sinC is a triangular calculation formula, so it is proved.

Certificate 4: Identity

Analysis: consider using S△ABC =r p, because there is the radius of the inscribed circle of a triangle, consider applying the identity of trigonometric function.

Identity: If ∠ A+∠ B+∠ C = 180χ, then

tg tg + tg tg + tg tg = 1

Proof: As shown in the figure, tg = ①.

tg = ②

tg = ③

According to this identity, we get:

+ + =

① ② ③ Substitute, and you get:

∴r2(x+y+z) = xyz ④

As shown in the figure, A+B-C = (X+Z)+(X+Y)-(Z+Y) = 2x.

∴x = in the same way: y = z =

Substituting into ④, we get: r 2 =

Multiply the two sides to get:

r 2 =

On both sides of the square, we get: r =

On the left is r = r p = s △ ABC, and on the right is the deformation of Helen formula, so it is proved.

Prove 5: Half Angle Theorem

Half-angle theorem: tg =

tg =

tg =

Proof: According to TG = ∴ R =× Y ①

Similarly, r =× z2r =× x③.

①× ②× ③, so r3 = ×xyz.