O is a point within the `∆ABC`. P, Q , R are three points on OA , OB and OC respectively such that `PQ||AB` and `QR||BC`. Prove that `RP||CA`.

In Fig., A, B and C are points on OP, OQ and OR respectively such that AB||PQ and AC||PR .Show that BC||QR .

O is a point in the interior of ∆ABC . OD,OE and OF are the perpendicular drawn to the sides BC, CA and AB respectively. Prove that AF^2+BD^2+CE^2=AE^2+BF^2+CD^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

D and E are points on the sides CA and CB respectively of a triangle ABC right angled at C.Prove that, AE^2+BD^2=AB^2+DE^2 .

ABC is a triangle. D and E are two points on AB and AC respectively such that DE is parallel to BC and AD=1 cm , BD= 2cm . What is the ratio of the area of /_\ABC to the area /_\ADE .

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

The bisector of angleC and angleG of two similar triangle ABC and EFG are meet the side AB and EF respectively at the point D and H.Prove that triangleDCA~triangleHGE

O is any point inside a rectangle ABCD. Prove that OB^2+OD^2=OA^2+OC^2 .

Two triangles ABC and DBC are in the same side of the common base BC. Lines drawn parallel to BA and BD from any point E on BC intersect AC and DC at the points P and Q respectively. Show that PQ is parallel to AD.

ABCD is a rectangle formed by the points A(-1,-1),B(-1,4),C(5,4) and D(5,-1).P,Q,R and S are the mid-points of AB,BC,CD and DA respectively.Is the quadrilateral PQRS a square?a rectangle?or a rhombus?Justify your answer.