figure:
or
. (Since the interior
angles of a quadrilateral sum to 360 degrees, each
quantity yields the same value for the cosine.)
Many texts in mathematics state Bretschneider’s
result without proof. Although algebraically detailed,
the derivation of the result is relatively straightforward.
One begins by noting that area K is the sum of the
areas of triangles ABC and ADC. We have:
Multiplying by four and squaring yields:
16K
2 = 4a
2 b
2 sin
2 B + 4c
2 d
2 sin
2 D + 8abcd sin B sin D
Call this equation (i). Applying the LAW OF COSINES to
each of the two triangles yields the relationship:
a
2 + b
2 – 2ab cos B = c
2 + d
2 – 2cd cos D
which can be rewritten:
a
2 + b
2 – c
2 – d
2 = 2ab cos B – 2cd cos D
Call this equation (ii). Also note that, with the aid of
this second equation:
4(s – c)(s – d) = (a + b – c + d)(a + b + c – d)
= a
2 + b
2 – c
2 – d
2 + 2ab + 2cd
= 2ab cos B – 2cd cos D + 2ab + 2cd
= 2ab(1 + cos B) + 2cd(1 – cos D)
Similarly:
4(s – a)(s – b) = (–a + b + c + d)(a – b + c + d)
= 2ab(1–cos B) + 2cd(1 + cos D)
Multiplying these two equations together gives:
16(s – a)(s – b)(s – c)(s – d)
= 4a
2
b
2
(1– cos
2 B) + 4c
2
d
2
(1– cos
2 D)
+ 4abcd((1 + cos B)(1 + cos D)
+ (1–cos B) (1–cos D))
= 4a
2
b
2 sin
2 B + 4c
2
d
2 sin
2 D +
8abcd(1 + cos B cos D)
and substituting back into equation (i) produces:
16K
2 = 16(s – a)(s – b)(s – c)(s – d) –
8abcd (cos B cos D – sin B sin D + 1)
The following identities from trigonometry:
cos B cos D – sin B sin D = cos(B + D)
and
now give:
which directly yields the famous result.
If, further, the opposite angles of the quadrilateral
sum to 180 degrees (in which case the quadrilateral is a
CYCLIC POLYGON), then Bretschneider’s formula reduces
to BRAHMAGUPTA’S FORMULA for the area of a cyclic
quadrilateral:
If one of the sides has length zero, say d = 0, then
the quadrilateral is a triangle and we have HERON’S
FORMULA for the area of a triangle:
This is valid, since every triangle can be inscribed in a
circle and so is indeed a cyclic polygon.
Briggs, Henry (1561–1630) British Logarithms Born
in February 1561 (the exact birth date is not known) in
Yorkshire, England, scholar Henry Briggs is remembered for the development of base-10 logarithms, revising the approach first taken by the inventor of
logarithms, JOHN NAPIER (1550–1617). Today such
common logarithms are sometimes called Briggsian logarithms. In 1617, after consulting with Napier, Briggs
published logarithmic values of the first 1,000 numbers
and, in 1624, in his famous text The Arithmetic of Logarithms, the logarithmic values of another 30,000 numbers, all correct to 14 decimal places.
K = √s(s – a)(s – b)(s – c)
K = √(s – a)(s – b)(s – c)(s – d)
16
16
16
2
2
2
K
s a s b s c s d
abcd
B D
=
−
−
−
− −
+
(
)(
)(
)(
)
cos
cos( )
cos
x
x
+ =
1 2
2
2
K
ab B
cd D
=
+
1
2
1
2
sin
sin
θ =
+
A C
2
θ =
+
B D
2
Briggs, Henry 53
or
. (Since the interior
angles of a quadrilateral sum to 360 degrees, each
quantity yields the same value for the cosine.)
Many texts in mathematics state Bretschneider’s
result without proof. Although algebraically detailed,
the derivation of the result is relatively straightforward.
One begins by noting that area K is the sum of the
areas of triangles ABC and ADC. We have:
Multiplying by four and squaring yields:
16K
2 = 4a
2 b
2 sin
2 B + 4c
2 d
2 sin
2 D + 8abcd sin B sin D
Call this equation (i). Applying the LAW OF COSINES to
each of the two triangles yields the relationship:
a
2 + b
2 – 2ab cos B = c
2 + d
2 – 2cd cos D
which can be rewritten:
a
2 + b
2 – c
2 – d
2 = 2ab cos B – 2cd cos D
Call this equation (ii). Also note that, with the aid of
this second equation:
4(s – c)(s – d) = (a + b – c + d)(a + b + c – d)
= a
2 + b
2 – c
2 – d
2 + 2ab + 2cd
= 2ab cos B – 2cd cos D + 2ab + 2cd
= 2ab(1 + cos B) + 2cd(1 – cos D)
Similarly:
4(s – a)(s – b) = (–a + b + c + d)(a – b + c + d)
= 2ab(1–cos B) + 2cd(1 + cos D)
Multiplying these two equations together gives:
16(s – a)(s – b)(s – c)(s – d)
= 4a
2
b
2
(1– cos
2 B) + 4c
2
d
2
(1– cos
2 D)
+ 4abcd((1 + cos B)(1 + cos D)
+ (1–cos B) (1–cos D))
= 4a
2
b
2 sin
2 B + 4c
2
d
2 sin
2 D +
8abcd(1 + cos B cos D)
and substituting back into equation (i) produces:
16K
2 = 16(s – a)(s – b)(s – c)(s – d) –
8abcd (cos B cos D – sin B sin D + 1)
The following identities from trigonometry:
cos B cos D – sin B sin D = cos(B + D)
and
now give:
which directly yields the famous result.
If, further, the opposite angles of the quadrilateral
sum to 180 degrees (in which case the quadrilateral is a
CYCLIC POLYGON), then Bretschneider’s formula reduces
to BRAHMAGUPTA’S FORMULA for the area of a cyclic
quadrilateral:
If one of the sides has length zero, say d = 0, then
the quadrilateral is a triangle and we have HERON’S
FORMULA for the area of a triangle:
This is valid, since every triangle can be inscribed in a
circle and so is indeed a cyclic polygon.
Briggs, Henry (1561–1630) British Logarithms Born
in February 1561 (the exact birth date is not known) in
Yorkshire, England, scholar Henry Briggs is remembered for the development of base-10 logarithms, revising the approach first taken by the inventor of
logarithms, JOHN NAPIER (1550–1617). Today such
common logarithms are sometimes called Briggsian logarithms. In 1617, after consulting with Napier, Briggs
published logarithmic values of the first 1,000 numbers
and, in 1624, in his famous text The Arithmetic of Logarithms, the logarithmic values of another 30,000 numbers, all correct to 14 decimal places.
K = √s(s – a)(s – b)(s – c)
K = √(s – a)(s – b)(s – c)(s – d)
16
16
16
2
2
2
K
s a s b s c s d
abcd
B D
=
−
−
−
− −
+
(
)(
)(
)(
)
cos
cos( )
cos
x
x
+ =
1 2
2
2
K
ab B
cd D
=
+
1
2
1
2
sin
sin
θ =
+
A C
2
θ =
+
B D
2
Briggs, Henry 53