Question

If D is a point on the side BC of ΔABC, then prove that
AB+BC+ CA>2AD.

Answer 1

Given: △ABC in which D is a point on BC
To prove: AB+BC+CA>2AD
Construction: AD is joined.
Proof: In △ABD,AB+BD>AD
[∵ the sum of any
two sides of a triangle isalways greater than the third side] --(1)

In △ADC,AC+DC>AD
[∵ the sum of any two sides of a triangle is always greater than the third side] ---- (2)

Adding (1) and (2) we get,
AB+BD+AC+DC>AD+AD
⇒AB+(BD+DC)+AC>2AD
⇒AB+BC+AC>2AD