It is known that quadrilateral ABCD is a quadrilateral inscribed in a circle, AB=BC=4, CD=2, and DA=6. Find the area of quadrilateral ABCD.
Here we use the generalization of Helen's formula.
S circle inscribed quadrilateral =
Under the radical symbol (p-a)(p-b)(p-c)(p-d).
(where P is the half circumference and A, B, C and D are the four sides)
Substituting into the solution results in s=8√.
three
[Edit this paragraph] Promotion
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, ha is the height of side A, and R and R are the radii of circumscribed circle and inscribed circle of △ABC, respectively, P.
=
(a+b+c)/2, then
S△ABC
= 1/2
aha
= 1/2
ab×sinC
= 1/2
r
p
=
2R2sinAsinBsinC
=
√[p(p-a)(p-b)(p-c)]
Where is S△ABC?
=√[p(p-a)(p-b)(p-c)]
It is the famous Helen formula, which was recorded by the Greek mathematician Helen in her book Geodesy.
[Edit this paragraph] Helen's formula has a very important application in solving problems. One,
Proof of Helen formula
justice
pythagorean theorem
As shown on the right
Pythagorean theorem proves Helen formula.
Certificate 2: Smith Theorem
As shown on the right.
The Proof of Smith Theorem and Helen Formula Ⅲ: Cosine Theorem
Analysis: from deformation ②
S
=
As we all know, by using cosine theorem.
c2
=
Aortic second sound
+
b2
-2abcosC
Prove it.
Prove: Prove S.
=
Prove
=
=
=
ab×sinC
At this time
=
Ab×sinC/2 is a triangular formula, so it is proved.
Certificate 4: Identity
Identification (1)
Proof of Identity (2) Proof of 5: Half Angle Theorem
∵ You Zheng Yi, X
=
=
-C.
=
Printed circuit board
y
=
=
[Ancient names or Latin modern names of animals and plants]
=
p-a
z
=
=
-B.
=
p-b
∴
r3
=
∴
r
=
∴S△ABC
=
r p
=
Therefore, it is proved that.
Second,
Generalization of Helen formula
Because it is often necessary to calculate the area of quadrilateral in practical application, it is necessary to popularize Helen formula. Since the triangle is inscribed in the circle, I guess Helen's formula can be summarized as follows: in any quadrilateral ABCD inscribed in the circle, let p=
S quadrilateral =
Now, according to the conjecture.
Proof: extend DA and CB to point e as shown in the figure.
Set EA
=
e
EB
=
f
∵∠ 1+∠2
= 180○
∠2+∠3
= 180○
∴∠ 1
=∠3
∴△EAB~△ECD
∴
=
=
=
Solution:
e
=
①
f
=
②
Due to the s-quadrilateral ABCD
=
EAB
Match ① and ② with B.
=
Substituting the formula variant ④, we get:
∴S quadrilateral ABCD
=
Thus the generalization of Helen's formula is proved.
[Edit this paragraph] Example:
As shown in the figure, the quadrilateral ABCD is inscribed in the circle O, while SABCD
=
, advertising
=
1，AB
=
1,
laser record
=
2.
Q: The quadrilateral may be an isosceles trapezoid.
Solution: let BC
=
x
From the popularization of Helen's formula, the following conclusions are drawn:
(4-x)(2+x)2
=27
x4- 12x2- 16x+27
=
x2(x2— 1)- 1 1x(x- 1)-27(x- 1)
=
(x- 1)(x3+x2- 1 1x-27)
=
x
=
1 or x3+x2- 1 1x-27
=
When x
=
1 year
=
B.C.
=
1
∴
The quadrilateral can be an isosceles trapezoid.
Implement in plan (VBS):
dimmed
a、b、c、p、q、s
A=inputbox ("Please enter the length of the first side of the triangle")
B=inputbox ("Please enter the length of the second side of the triangle")
C=inputbox ("Please enter the length of the third side of the triangle")
a= 1*a
b= 1*b
c= 1*c
p =(a+b+c)*(a+b-c)*(a-b+c)*(-a+b+c)
q=sqr(p)
s=( 1/4)*q
Msgbox ("triangle area is"&; s)，
, "triangle area"
Implemented in VC
# include & ltstdio.h & gt
# include & ltmath.h & gt
Master ()
{
Internationalorganizations (same as international organizations)
a、b、c、s；
Printf ("Enter the first side \ n");
scanf("%d "，& ampa)；
Printf ("Enter the second side \ n");
scanf("%d "，& ampb)；
Printf ("Enter the third side \ n");
scanf("%d "，& ampc)；
s =(a+b+c)/2；
Printf ("area: %f\n "，sqrt(s *(s-a)*(s-b)*(s-c))；
}
Heron's formula