P and Q are the mid-points of the sides CA and CB respectively of a `Delta ABC`, right angled at C. Prove that `4(AQ^(2)+BP^(2))=5 AB^(2)`.

BL and CM are medians of a triangle ABC right angled at A. Prove that 4(BL^(2)+ CM^(2))=5BC^(2)

ABC is an isosceles triangle right angled at C. Prove that AB^(2) = 2AC^(2) .

If D,E and F are the mid-points of the sides BC, CA and AB respectively of a triangle ABC and lambda is scalar, such that vec(AD) + 2/3vec(BE)+1/3vec(CF)=lambdavec(AC) , then lambda is equal to

In a right triangle ABC right angled at C, P and Q are points on sides AC and CB respectively which divide these sides in the ratio of 2 : 1. Prove that (i) 9 AQ^(2) = 9AC^(2) + 4BC^(2) (ii) 9BP^(2) = 9BC^(2) + 4AC^(2) (iii) 9 (AQ^(2) + BP^(2)) = 13 AB^(2)

If D and E, are the midpoints of the sides AB and AC of a triangle ABC, prove that vec(BE) +vec(DC)=(3)/(2)vec(BC).

In Delta ABC, angle BAC = 90^(@), seg BL and seg CM are medians of Delta ABC prove that 4(BL^(2) + CM^(2)) = 5 BC^(2)

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

D, E, F are mid points of sides BC, CA, AB of Delta ABC . Find the ratio of areas of Delta DEF and Delta ABC .