How to find a corner between diagonals

How to find a corner between diagonals

Polygon diagonal - a piece which connects two figure tops which are not adjoining among themselves (i.e. non-adjacent tops or not belonging to one party of a polygon). In a parallelogram, knowing length of diagonals and length of the parties, it is possible to calculate corners between diagonals.


1. For convenience of perception of information draw any parallelogram of ABCD on the sheet of paper (the parallelogram is a quadrangle which opposite sides are in pairs equal and parallel). Connect opposite tops pieces. Received the EXPERT and BD – diagonals. Designate a point of intersection of diagonals by letter O. It is necessary to find corners of BOC (AOD) and COD (AOB).

2. The parallelogram has a number of mathematical properties: - diagonals are halved by a point of intersection; - diagonal parallelogram divides it into two equal triangles; - the sum of all corners is equal in a parallelogram 360 degrees; - the sum of the corners adjacent to one party of a parallelogram is equal to 180 degrees; - the sum of squares of diagonals is equal to the double sum of squares of its adjacent parties.

3. To find corners between diagonals, use the theorem of cosines from the theory of elementary geometry (Euclidean). According to the theorem of cosines, the square of the party of a triangle (A) can be received, having put squares of two of its other parties (B and C) and to subtract the double work of these parties from the received sum (B and C) on a cosine of the angle between them.

4. In relation to ABCD parallelogram VOS triangle the theorem of cosines will look as follows: VS square = a square IN + OS square – 2*BO*OC*cos of corner VOSOtsyuda cos of a corner of BOC = (VS square – a square IN – OS square) / (2*BO*OC)

5. Having found value of a corner of BOC (AOD) it is easy to calculate value of other corner concluded between diagonals – COD (AOB). For this purpose subtract value of a corner of BOC (AOD) from 180 degrees – since the sum of adjacent corners is equal to 180 degrees, and corners of BOC and COD and corners of AOD and AOB – adjacent.

Author: «MirrorInfo» Dream Team