In Fig. 4.220, D is the mid-point of side BC and . If BC = a, AC = b, AB = c, ED = x, AD = p and AE = h, prove that:

(i) (ii)

(iii)

We have

D is the midpoint of BC

(i) In AEC

AC²=AE²+EC²

b²=AE²+(ED+DC)²

b²=AD²+DC²+2xEDxDC (Given BC=2CD)

b²=p²+(a/2)²+2(a/2)x

b²=p²+a²/4+ax

b²=p² +ax+a²/4 ………….. (i)

(ii) In AEB

AB²=AE²+BE²

c²=AD²-ED²+(BD-ED)²

c²=p²-ED²+BD²+ED²-2BDxED

c²=P²+(a/2)²-2(a/2)²x

c²=p²-ax+a²/4 ……………….(ii)

(iii) Adding equ. (i)and(ii) we get

b²+c²=2p²+a²/2