E is the middle point of the median AD of `/_\ABC` . BE intersects AC at F. Prove that `AF=1/3AC`.

E is the middle point of the median AD of /_\ABC . BE produced meets AC at F. Prove that CA=3AF .

ABCD is a parallelogram. E is the middle point of the side CD. BE intersects AC at the point X. prove that AX=2/3AC .

E is a point on the side AD produced of a parallelogram ABCD and BE intersects CD at F. Show that triangleABE~triangleCFB .

AB and CD two parallel sides of trapezium ABCD where AB=2 CD . AC and BD intersects at X. prove that AX=2/3AC .

If D is the middle point of the side BC of the triangle ABC, prove that AB^2 + AC^2 = 2(AD^2 + DC^2) .

The right bisector of AB and AC of the triangle ABC intersect at 0. Prove that OA = OB = OC .

The perpendicular from C on the side AB of a triangle ABC intersect AB at D and BD=mAD . Prove that (m+1)BC^2=(m+1)AC^2+(m-1)AB^2 .

If x and y are the mid-points of the sides CA and CB respectively of a ∆ABC right angled at C. prove that 4Ay^2=4AC^2 +BC^2

AD is the bisector of /_BAC of /_\ABC , where D is a point on BC. Prove that (BD)/(DC)=(AB)/(AC) .

ABC and DBC are two right angled triangles with common hypotenuse BC with their sides AC and BD intersecting at P. Prove that: AP×PC=DP×PB .