Question

In the quadrilateral abcd , prove that a b + bc + cd + d a greater than ac + bd

Answer 1

ABCD is a quad.Its diagonals are AC and BD.

In triangle ACB, AB + BC > AC - 1

In triangle BDC, BC + CD > BD - 2

In triangle ACD, AD + DC > AC -3

In triangle BAD, AB + AD > BD -4

Adding 1,2,3 and 4,

AB + BC + BC + CD + AD + DC + AB + AD > AC + BD + AC + BD

2AB + 2BC + 2CD + 2AD > 2AC + 2BD

AB + BC + CD + AD > AC + BD.

Answer 2

Answer:

ABCD is a quad.Its diagonals are AC and BD.

In triangle ACB, AB + BC > AC  ......................(1)

In triangle BDC, BC + CD > BD .....................(2)

In triangle ACD, AD + DC > AC ...................(3)

In triangle BAD, AB + AD > BD ....................(4)

By Adding (1) (2) (3) and (4),

AB + BC + BC + CD + AD + DC + AB + AD > AC + BD + AC + BD

2AB + 2BC + 2CD + 2AD > 2AC + 2BD

AB + BC + CD + AD > AC + BD.