First part :


PVS and UVQ are similar (as proved in bi)

PS/UQ = SV/QV = 2/1

(PS/UQ = 2/1 because :

PS = QR (opposite sides of a parallelogram are equal in length)

Since U is the midpoint of QR,

UQ = UR

Then,

QR = UQ + UR
= UQ + UQ
= 2UQ

Since PS = QR, then QR = 2UQ )

Since SV/QV = 2/1 ,

SV = 2QV

Since SQ = SV + QV,

SQ = 2QV + QV = 3QV

Which means QV = ⅓ SQ

Now, T is the intersection of both diagonals. It is also their midpoint.


So ST = QT = ½ SQ

VT = QT - QV

= ½ SQ - ⅓ SQ

= 1/6 SQ


Lastly,

QV/VT = ⅓SQ / 1/6SQ
= 2/1

So QV = 2VT (proved)