O is any point inside a triangle ABC. The bisector of `angle AOB, angle BOC and angle COA` meet the sides AB, BC and CA in point D, E and F respectively.
Show that `AD xx BE xx CF= DB xx EC xx FA`

ABCD is a quadrilateral in which AB= AD, the bisector of angle BAC and angle CAD intersect the sides BC and CD at the points E and F respectively. Prove that EF || BD.

Let P be an interior point of a triangle ABC and AP, BP, CP meet the sides BC, CA, AB in D, E, F, respectively. Show that (AP)/(PD)=(AF)/(FB)+(AE)/(EC) .

In a triangle ABC, if D and E are mid points of sides AB and AC respectively. Show that vecBE+ vecDC=(3)/(2)vecBC .

In Fig. the incircle of △ABC, touches the sides BC, CA and AB at D, E respectively. Show that : AF + BD + CE = AE + BF + CD = ½ (Perimeter of △ABC).

let P an interioer point of a triangle A B C and A P ,B P ,C P meets the sides B C ,C A ,A BinD ,E ,F , respectively, Show that (A P)/(P D)=(A F)/(F B)+(A E)/(E C)dot

In a Delta ABC , AD is the bisector of angle A meeting side BC at D, if AB=10 cm, AC =14 cm and BC=6 cm, find BD and DC.

ABC is right - angled triangle at B. Let D and E be any two point on AB and BC respectively . Prove that AE^2 + CD^2 = AC^2 + DE^2

Let A, B, and C be the vertices of a triangle. Let D, E, and F be the midpoints of the sides BC, CA, and AB respectively. Show that vec(AD)+vec(BE)+vec(CF)=vec(0) .

From a point O inside a triangle ABC, perpendiculars OD, OE and OF are drawn to the sides BC, CA and AB, respectively. Prove that the perpendiculars from A, B and C to the sides EF, FD and DE are concurrent

Draw a triangle ABC of base BC=8 cm, angle A=60^(@) and the bisector of angle A meets BC at D such that BD=6 cm.