Lex X by any point on the side BC of a triangle ABC. If XM, XN are drawn parallel to BA and CA meeting CA, BA in M, N respectively; MN meets BC produced in T, prove that `T X^2=T B` x `T C`

ABC is a triangle A line is drawn parallel to BC to meet AB and AC in D and E respectively. Prove that the median through A bisects DE.

D and F are the mid-points of sides AB and AC of a triangle ABC. A line through F and parallel to AB meets BC at point E. Prove that BDFE is a parallelogram

XY is a line parallel to side BC of a triangle ABC . If B E ||A C and C F\ ||\ A B meet XY at E and F respectively, show that a r\ (A B E)\ =\ a r\ (A C F)

The side B C of a triangle A B C is bisected at D ; O is any point in A D. BO and CO produced meet AC and AB in E and F respectively and A D is produced to X so that D is the mid-point of O X . Prove that A O : A X=A F : A B and show that FE||BC .

X and Y are points on the side LN of the triangle LMN such that LX = XY = YN. Through X, a line is drawn parallel to LM to meet MN at Z (see figure). Prove that ar (DeltaLZY) = ar (MZYX) .

The diameters of the circumcirle of triangle ABC drawn from A,B and C meet BC, CA and AB, respectively, in L,M and N. Prove that (1)/(AL) + (1)/(BM) + (1)/(CN) = (2)/(R)

In triangle ABC, D and E are points on side AB such that AD = DE = EB. Through D and E, lines are drawn parallel to BC which meet side AC at points F and G respectively. Through F and Glines are drawn parallel to AB which meet side BC at points M and N respectively. Prove that : BM = MN = NC.

O is any point in the plane of the triangle ABC,AO,BO and CO meet the sides BC,CA nd AB in D,E,F respectively show that (OD)/(AD)+(OE)/(BE)+(OF)/(CF)=1.

In Delta ABC the lines are drawn parallel to BC,CA and AB respectively through A,B and C intersecting at P, Q and R. Find the ration of perimeter of Delta PQR and DeltaABC

In the adjoining figure, DeltaPQR is formed by the sides PQ, QR and RP which are drawn parallel to sides AB,BC and CA respectively of DeltaABC. Prove that " "PQ+QR+RP=2(AB+BC+CA).