A line L intersects three sides BC, CA and AB of a triangle in P,Q,R respectively, show that `(BP)/(PC).(CQ)/(QA).(AR)/(RB)=-1`

A(x_1,y_1), B(x_2,y_2),C(x_3,y_3) are three vertices of a triangle ABC, lx+my+n=0 is an equation of line L. If L intersects the sides BC,CA and AB of a triangle ABC at P,Q,R respectively, then (BP)/(PC)xx(CQ)/(QA)xx(AR)/(RB) is equal to

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.

P and Q are respectively the mid-points of sides AB and BC of a triangle ABC and Ris the mid-point of AP, show that (i) a r (P R Q)=1/2a r(A R C) (ii) a r (R Q C)=3/8a r (A B C) (iii) a r (P B Q) =a r (ARC)

In the following figure, OP, OQ and OR are drawn perpendiculars to the sides BC, CA and AB repectively of triangle ABC. Prove that : AR^(2)+BP^(2)+CQ^(2)=AQ^(2)+CP^(2)+BR^(2)

Let A, B, C respresent the vertices of a triangle, where A is the origin and B and C have position b and c respectively. Points M, N and P are taken on sides AB, BC and CA respectively, such that (AM)/(AB)=(BN)/(BC)=(CP)/(CA)=alpha Q. AN+BP+CM is

Through A, B and C lines RQ, PR and QP have been drawn, respectively parallel to sides BC, CA and AB of a Delta ABC as shown in figure. Show that BC=(1)/(2)QR .

Let A, B, C represent the vertices of a triangle, where A is the origin and B and C have position b and c respectively.* Points M, N and P are taken on sides AB, BC and CA respectively, such that (AM)/(AB)=(BN)/(BC)=(CP)/(CA)=alpha . If triangle represent the area enclosed by the three vectors AN, BP and CM, then the value of alpha , for which triangle is least

The midpoints of the sides BC, CA and AB of a triangle ABC are D(2,1), E(-5,7)and F(-5, -5) respectively. Find the equations of the sides of triangle ABC .

In the following diagram, lines l, m and n are parallel to each other. Two transversals p and q intersect the parallel lines at points A, B, C and P, Q, R as shown. Prove that : (AB)/(BC) = (PQ)/(QR) Join A and R. Let AR meets BQ at point D.

In a triangle ABC, the incircle (centre O) touches BC , CA and AB at points P,Q and R respectively. Calculate: (i) /_QOR (ii) /_QPR given that /_A=60^(@)