In Figure altitudes AD and CE of DABC intersect each other at the point P. Show that: (i) `DeltaA E P DeltaC D P` (ii) `DeltaA B D DeltaC B E` (iii) `DeltaA E P DeltaA D B` (iv) `DeltaP D C DeltaB E C`

In Figure, two chords AB and CD intersect each other at the point P. Prove that: (i) DeltaA P C~ DeltaD P B (ii) A P*P B=C P*D P

In Figure two chords AB and CD of a circle intersect each other at the point P (when produced) outside the circle. Prove that (i) DeltaP A C DeltaP D B (ii) P AdotP B=P CdotP D

In figure, if DeltaA B E~=DeltaA C D , show that DeltaA D E~ DeltaA B C .

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that DeltaA B E DeltaC F B .

DeltaA B C and DeltaD B C are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC (see Fig. 7.39). If AD is extended to intersect BC at P, show that (i) \ DeltaA B D~=DeltaA C D (ii) DeltaA B P~=DeltaACP (iii) AP bisects ∠ A as well as ∠ D (iv) AP is the perpendicular bisector of BC

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 figure Cm and RN are respectively the medians of DeltaA B C and DeltaP Q R . If DeltaA B C ~DeltaP Q R , prove that: (i) DeltaA M C ~DeltaP N R (ii) (C M)/(R N)=(A B)/(P Q) (ii) DeltaC M B ~DeltaR N Q

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 quadrilateral ACBD, A C\ =\ A D and AB bisects /_A (see Fig. 7.16). Show that DeltaA B C~=DeltaA B D

In Fig. 4.101, A D and C E are two altitudes of A B C . Prove that A E F ~ C D F (ii) A B D ~ C B E (iii) A E F ~ A D B (iv) F D C ~ B E C (FIGURE)