ABC is a triangle right angled at C. A line through the midpoint M of hypotenuse AB and Parallel to BC intersects AC at D. Show that
(i) D is the midpoint of AC
(ii) `MD_|_AC`
(iii) `CM=MA=(1)/(2)AB.`

In trapezium ABCD, AB || CD and E is the midpoint of AD. A line drawn through E and parallel to AB intersects BC at F. Prove that F is the midpoint of BC and EF = (1)/(2)(AB + CD) .

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

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

ABC is a right triangle right angled at B. Let D and E be any points on AB and BC respectively. Prove that AE^(2) + CD^(2) = AC^(2) + DE^(2) .

In right triangle ABC, right angled at C, M is the midpoint of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see the given figure). Show that: (i) triangle AMC = triangleBMD (ii) angle DBC is a right angle (iii) triangleDBC = triangleACB (iv) CM=1/2 AB

In Delta ABC , AD is a median and E is the midpoint of AD. BE is extended to intersect AC at F. Prove that AF = (1)/(3)AC .

In right triangle ABC, right angle is at C, M is the mid-point of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B (see figure). Show that : (i) DeltaAMC ~= DeltaBMD (ii) /_DBC is a right angle (iii) Delta DBC ~= DeltaACB (iv) CM = 1/2 AB .

In triangle ABC, the bisectors of angle B and angle C intersect at I. A line drawn through I and parallel to BC intersects AB at P and AC at Q. Prove that PQ = BP + CQ.

In the given figure, ABD is a Deltaright angled at A and AC bot BD . Show that AB^(2)= BC*BD

ABC is a right angled triangle in which angle A = 90^(@) and AB = AC. Find angle B and angle C.