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

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

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 .

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) .

D is the point on the side BC of ABC such that angleADC = angleBAC . Prove that, (BC)/(CA) = (CA)/(CD) .

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) .

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

P and Q are points on the sides CA and CB of a ∆ABC , right angled at C. Prove that AQ^2 + BP^2 = AB^2 + PQ^2 .

A circle is touching the side BC of /_\ABC at P and touching AB and AC produced at Q and R are respectively. Prove that AQ=1/2(AB+BC+CA)=1/2(Perimeter of /_\ABC) .

If PB and AQ are the medians of a ∆ABC right angled at C. Prove that 4AQ^2 = 4 AC^2 + BC^2