correct Blaschke formula (Re: [EMHL] Area of a quadrilateral)
- Hello Marcello,
The formula you are asking for is the following. Let ABCD be (not
nesessary convex) quadrilateral with sides AB=k, BC=l, CD=m, DA=n and
angles 2a, 2b, 2c, 2d at vertices A,B,C,D respectively; and let S be the
area of the quadrilateral.
4*S = (k+l+m+n)^2/(cot(a)+cot(b)+cot(c)+cot(d)) -
As far as I know, this formula was first discovered by german
mathematition Wilhelm Blaschke. First appeared in 1914 in:
Jahresbericht Deutsch. Mathem. Vereinigung 23, 1914, page 210-234
Plain geometry proof and several computer-based proofs of this formula can
be found in the article:
"Russian killer No. 2, A challenging geometric theorem with human and
machine proofs" by Xiaorong Hou, Hongbo Li, Dongming Wang and Lu Yang
Mathematical Intelligencer, Volume 23, number 1, 2001, pages: 9-15
It can be downloaded from
Size of file = 1 387 554 bytes, PDF version 1.2 (so you will need Adobe
Acrobat Reader version 3.0 or later).
- Dear Sergei,
Since you were the one who sent the Intelligencer article authors the 5
can you post the 5 problems here please?
Date: Thu Jun 7, 2001 11:07 am
Subject: 5 Russian killers
A recent issue of Mathematical Intelligencer* contains an
article by 4 Chinese authors on Number 2 of 5 Russian killers.
wrote that ``these theorems have been used to prepare the
for the all-Russian school mathematics olympiad, and are called
killers to analytic ways of geometric problem-solving''. Here is
Theorem. Let ABCD be an arbitrary quadrilateral with sides AB
= k, BC
= ell, CD = m, DA = n, and internal angles 2a, 2b, 2c, 2d at
A, B, C, D respectively; and let S be the area of the
4S = (k+ell+m+n)^2/F - (k-ell+m-n)^2/G,
F = cot a + cot b + cot c + cot d,
G = tan a + tan b + tan c + tan d.
Does any one know what the other killers are?
*volume 23 (2001) 9 -- 15.