In parallelogram ABCD two points P and Q are taken on diagonal BD such that DP = BQ (set figure). Show that:
`{:((i)DeltaAPD~=CQB,(ii)AP=CQ),((iii)DeltaAQB~=DeltaCPD,(iv)AQ=CP),((v)APCQ "is a parallelogram.",):}` ltbegt

In parallelogram ABCD, two points P and Q are taken on diagonal BD such that D P = B Q . Show that: (i) DeltaA P D~= DeltaC Q B (ii) A P = C Q (iii) DeltaA B C (iv) A Q = C P (v) APCQ is a parallelogram.

In a parallelogram ABCD,points P and Q are points of trisection of diagonal BD.Prove that CQ is parallel to AP.

ABCD is a parallelogram and AP and CQ are perpendiculars from vertices A and C on diagonal BD . Show that (i) DeltaA P B~=\ DeltaC Q D (ii) A P\ =\ C Q

In the given figure, ABCD is a parallelogram in which AN and CP are perpendiculars on diagonal BD. Prove that : (i) DeltaADN =DeltaCBP (ii) AN=CP

In the figure, ABCD is a parallelogram and BC is produced to a point Q such that AD = CQ. If AQ intersect DC at P, show that ar( Delta BPC) = ar( Delta DPQ).

ABCD is a parallelogram. Tow points P and Q are taken on sides AD and BC respectively such that AP 1/3ADand CQ=1/3BC. Prove that squareAQCP is a parallelogram.

In the adjoining figure, P and Q are two points on equal sides AB and AC of an isosceles triangle ABC such that AP = AQ. Prove that BQ = CP.

ABCD is a parallelogram in which P and Q are midpoints of opposite sides AB and CD respectively (see figure). If AQ intersects DP at S and BQ intersects CP at R, show that: (i) APCQ is a parallelogram.

ABCD is a parallelogram in which P and Q are mid-points of opposite sides AB and CD (see Fig. 8.18). If AQ intersects DP at S and BQ intersects CP at R, show that: (i) APCQ is a parallelogram. (ii) DPBQ is a parallelogram. (iii) PSQR is a parallelogram

DeltaABC and DeltaDBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see figure). If AD is extended to intersect BC at P, show that : (i) DeltaABD cong DeltaACD (ii) DeltaABP cong DeltaACP (iii) AP bisects angleA as well as angleD (iv) AP is the perpendicular bisector of BC.