# Bretschneider's formula

Bretschneider's formula gives the area of a quadrilateral, \(\Delta\), by the following formula. \[\Delta^{2} = (s-a)(s-b)(s-c)(s-d)-abcd\cos^{2}\left(\frac{B+D}{2}\right)\]

Notation: Let \([ABCD]\) represent the area of \(\Box ABCD\) and similar for other triangles. Let \(\Delta = \Box ABCD\), and let \(AB=a\), \(BC=b\), \(CD=c\), \(DA=d\). Let \(s\) be the semiperimeter of \(\Box ABCD\), given by \(s=\frac{a+b+c+d}{2}\).

Construction: Join \(\overline{AC}\).

Proof:

\([ABCD] = [ABC] + [ACD]\)

\(\therefore \Delta = \frac{1}{2}ab\sin B + \frac{1}{2}cd\sin D\)

\(\therefore 2ab\sin B + 2cd\sin D = 4\Delta\)...................(1)

Now we will use cosine rule to find \(AC\) in two ways, once using \(\Delta BAC\) and once using \(\Delta DAC\).

\(AC^{2} = a^{2} + b^{2} - 2ab\cos B = c^{2} + d^{2} - 2cd\cos D\)

\(\therefore 2ab\cos B - 2cd\cos D = a^{2} + b^{2} - c^{2} - d^{2}\)..........................(2)

Squaring and adding equations (1) and (2),

\(4a^{2}b^{2}+4c^{2}d^{2} - 8abcd(\cos B \cos D - \sin B \sin D) = 16\Delta^{2} + (a^{2}+b^{2} - c^{2}-d^{2})^{2}\)

\(4a^{2}b^{2}+4c^{2}d^{2}-8abcd\cos(B+D)=16\Delta^{2}+(a^{2}+b^{2}-c^{2}-d^{2})^{2}\)

\((2ab+2cd)^{2}-8abcd-8abcd\cos(B+D)=16\Delta^{2}+(a^{2}+b^{2}-c^{2}-d^{2})^{2}\)

\((2ab+2cd)^{2}-8abcd(1+\cos(B+D))=16\Delta^{2}+(a^{2}+b^{2}-c^{2}-d^{2})^{2}\)

\(16\Delta^{2}=(2ab+2cd)^{2}-(a^{2}+b^{2}-c^{2}-d^{2})^{2}-8abcd\times 2\cos^{2}\left(\frac{B+D}{2}\right)\)

\(=(2ab+2cd+a^{2}+b^{2}-c^{2}-d^{2})\times(2ab+2cd-a^{2} - b^{2}+c^{2}+d^{2})-16abcd\cos^{2}\left(\frac{B+D}{2}\right)\)

\(=((a+b)^{2}-(c-d)^{2})((c+d)^{2}-(a-b)^{2})-16abcd\cos^{2}\left(\frac{B+D}{2}\right)\)

\(=(a+b+c-d)(a+b-c+d)(a-b+c+d)(-a+b+c+d)-16abcd\cos^{2}\left(\frac{B+D}{2}\right)\)

\(=(2s-2d)(2s-2c)(2s-2b)(2s-2a)-16abcd\cos^{2}\left(\frac{B+D}{2}\right)\)

\(16\Delta^{2}=16(s-a)(s-b)(s-c)(s-d)-16abcd\cos^{2}\left(\frac{B+D}{2}\right)\)

\(\Delta^{2}=(s-a)(s-b)(s-c)(s-d)-abcd\cos^{2}\left(\frac{B+D}{2}\right)\)

This is Bretschneider's formula.

Also see the pages on cyclic quadrilaterals and Brahmagupta's formula.

**Cite as:**Bretschneider's formula.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/bretschneiders-formula/