In a right ` A B C` right-angled at `C` , if `D` is the mid-point of `B C` , prove that `B C^2=4\ (A D^2-A C^2)` .

In right-angled triangle A B C in which /_C=90^@ , if D is the mid-point of B C , prove that A B^2=4\ A D^2-3\ A C^2 .

In fig., if A D_|_B C , prove that A B^2+C D^2=B D^2+A C^2 .

In a triangle A B C ,\ A C > A B , D is the mid-point of B C and A E_|_B C . Prove that: (i) A B^2=A D^2-B C* D E+1/4B C^2

If A B C is a right triangle right-angled at Ba n dM ,N are the mid-points of A Ba n dB C respectively, then 4(A N^2+C M^2)= (A) 4A C^2 (B) 5A C^2 (C) 5/4A C^2 (D) 6A C^2

In a triangle A B C ,\ A C > A B , D is the mid-point of B C and A E_|_B C . Prove that: (i) A B^2=A D^2-B C . D E+1/4B C^2 (ii) A B^2+A C^2=2\ A D^2+1/2B C^2

In right triangle A B C , right angle at C ,\ M is the mid-point of the hypotenuse A Bdot C is jointed to M and produced to a point D such that D M=C Mdot Point D is joined to point Bdot Show that A M C~= B M D (ii) /_D B C=/_A C B D B C~= A C B (iii) C M=1/2A B

In a right triangle A B C right-angled at B , if P a n d Q are points on the sides A Ba n dA C respectively, then (a) A Q^2+C P^2=2(A C^2+P Q^2) (b) 2(A Q^2+C P^2)=A C^2+P Q^2 (c) A Q^2+C P^2=A C^2+P Q^2 (d) A Q+C P=1/2(A C+P Q)dot

In A B C , A D is a median. Prove that A B^2+A C^2=2\ A D^2+2\ D C^2 .

A B C is a triangle right angled at B ; a n d\ P is the mid-point of A C . Prove that: P Q\ _|_A B (ii) Q is the mid point of A B P B=P A=1/2A C

In Figure, A B C is a right triangle right angled at B and points Da n dE trisect B C . Prove that 8A E^2=3A C^2+5A D^2dot