P is any point on BC of the square ABCD. The perpendicular , drawn from B to AP intersects DC at Q . Prove that AP = BQ.

I is the incentre of the Delta ABC . The perpendicular drawn from I to BC intersects BC at. P. Prove that AB-AC=BP-CP .

D is the mid-point of the side BC of the DeltaABC and P is any point on BC. Let us join P, A. The line segment drawn through D, parallel to PA intersects AB at Q. Prove that DeltaBPQ=1/2DeltaABC .

D and E are the mid-points of AB and AC respectively of the DeltaABC. AP is the median of BC. AP intersects DE at Q. Prove that DQ = QE and AQ = QP.

P is any point on diameter of a circle with centre O.A perpendicular drawn on diameter at the point O intersects the point O intersects the circle at the point Q. Extended QP intersects the circle at the point R.A tangent drawn at the point R intersects extended OP at the point S. Prove theat SP=SR. [GP-X]

D is any point on the side BC of the DeltaABC . P and Q are the centroids of DeltaABD and DeltaACD . Prove that PQ||BC .

ABCD is a cyclic quadrilaterla. Extended AB and DC intersect at P. Prove that PA.PB = PC.PD.

M is such a point on AD of the square ABCD that angleCMD = 60^(@) . The diagonal BD intersect CM at the point P , then find angleDPC .

M is a point on the side AD of the square ABCD such that angleCMD=30^(@) . If the diagonal BD intersects CM at a point P , then find angleDPC .

Modhurima have drawn a semicircle with a diameter AB . A perpendicular is drawn on AB from any point C on AB which intersects the semicircle at the point D . Prove that CD is a mean proportion of AC and BC .

In DeltaABC , AB = AC. The perpendiculars drawn from B and C to AC and AB respectively, intersect the sides AC and AB at the points E and F respectively. Prove that FE||BC.