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 `

In the adjacent figure, ABC is a right angled triangle with right angle at B and points D, E trisect BC. Prove that 8AE^(2)=3AC^(2)+5AD^(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 .

ABC is a right triangle , right angled at A and D is the mid point of AB . Prove that BC^(2) =CD^2 +3BD^(2) .

In the figure, ABC is a triangle in which AB= AC. Points D and E are points on the side AB and AC respectively such that AD=AE. Show that points B, C, E and D lie on a same circle.

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)

In an isosceles triangle ABC with AB = AC, D and E are points on BC such that BE = CD (see figure) Show that AD = AE

ABCD is a square. E, F, G and H are the mid points of AB, BC, CD and DA respectively. Such that AE = BF = CG = DH . Prove that EFGH is a square.

In the given fig. if AD bot BC Prove that AB^(2) + CD^(2) = BD^(2) + AC^(2) .