P and Q re tow interior points on te side BC of `/_\ABC `such that , `BP ||BQ` and `BC.PQ=BP. CQ` and AQ bisects `/_PAC` using vector method prove that AQ and AB are mutually perpendicular

P and Q are the points on sides AB and AC respectively of a DeltaABC , such that PQ||BC .If AP|PB=2:3 and AQ=4 cm , then the length of AC is ….. Cm .

P and Q are points on the sides CA and CB of a ∆ABC , right angled at C. Prove that AQ^2 + BP^2 = AB^2 + PQ^2 .

D is the midpoint of the side BC of Delta ABC . If P and Q are points o AB and on AC such that DP bisects angle BDA and DQ bisects angle ADC , then prove that PQ || BC

In a Delta ABC , let P and Q be points on AB and AC respectively such that PQ || BC . Prove that the median AD bisects PQ.

P and Q are points on the sides AB and AC of a ABC such that BP = CQ = x cm, PA = 6cm, AQ = 20cm, BC = 25cm. If PAQ and quadrilateral BPQC have equal areas, then the value of x is (A) 2 cm (B) 3 cm (C) 4 cm (D) 5 cm

Let ABC be a triangle with circumcenre O. The points P and Q are interior points of the sides Let K, L and M be the mid points of the segments BP, CQ and PQ respectively, and let tau be the circle passing through K, L and M. Suppose that the line PQ is tangent to the circle tau. Prove that OP=OQ.

In the below given figure P and Q are points on the sides AB and AC respectively of triangle ABC such that AQ = 3cm, QC= 5cm and PQ ||BC . Find the ratio of areas traingle APQ and triangle ABC .