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.

O is any point inside a triangle ABC. The bisector of /_AOB,/_BOC and /_COA meet the sides AB,BC and CA in point D, and F respectively.Show that ADxBExCF =DBxECFA

Let's denote the semi-perimeter of a triangle ABC in which BC = a, CA = 5, AB = c. If a circle touches the side BC, CA, AB at D, E, F respectively, prove that BD = S-b.

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.

The side BC of a triangle ABC is bisected at D;o is any point in AD,BO and CO produced meet AC and AB in E and F respectively and AD is produced to X so that D is the mid-point of OX. Prove that AO:AX=AF:AB and show that FE|BC.

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