Question

Prove that the sum of the length of the diagonals of a quadrilateral is less than the perimeter, but greater than the half of the perimeter of this quadrilateral. Using statement reason format.

Answer 1

See attached image.

(1) Prove that sum of lengths of the diagonals A and B is less that the perimeter. I am going to call A and B the full diagonal (A=A1+A2, B=B1+B2) for this part:

Use triangle inequalities:

A < a+b

A < c + d

B < c + b

B < a + d

add them up

A + A + B + B < a+b+c+d+c+b+a+d

2(A + B) < 2(a+b+c+d)

(sum of doagonals) A+B < a+b+c+d (perimeter) Q.E.D

(2) Prove that sum of diagonals is greater than half perimeter:

use triangle inequality:

A1+B1>a

B1+A2>b

A2+B2>c

B2+A1>d

add them up

2(A1+A2+B1+B2) > a+b+c+d

(sum of diagonals) A+B > 0.5(a+b+c+d) (half perimeter) Q.E.D