We know that AR || BC and AB || RC
From the figure we know that ABCR is a parallelogram
So we get
AR = BC …… (1)
We know that AQ || BC and QB || AC
From the figure we know that AQBC is a parallelogram
So we get
QA = BC ……… (2)
By adding both the equations
AR + QA = BC + BC
We know that AR + QA = QR
So we get
QR = 2BC
Dividing by 2
BC = QR/2
BC = ½ QR
Therefore, it is proved that BC = ½ QR.