`DeltaABC` is inscribed in a circle, `AD` is a diameter of the circle and `AE` is perpendicular on the side `BC`, which intersects the side `BC` at the point `E`. Prove that `DeltaAEB` and `DeltaACD` are similar. From this also prove that `ABxxAC=AExxAD`.

AD is a diameter of the circumcircle of DeltaABC . AE is perpendicular to BC . Prove that ABxxAC=ADxxAE .

O is the circumcentre of Delta ABC and OD is perpendicular on the side BC, prove that angle BOD= angle BAC

If D is the mid point of side BC of a triangle ABCand AD is perpendicular to BC, then

Delta ABC is an acute angle triangle inscribed in a circle in a circle .AD is a diameter of the circle . Two perpendiculars BE and CF are drawn from B and C to AC and AB respectively , which intersect each other at the point G . Prove that BDCG is a parallelogram.

The two chords PQ and RS of a circle intersects each other at the point X within the circle. By joining P,S and R,Q prove that DeltaPXS and DeltaRSQ are similar. From this also prove that PX: XQ=RX : XS .

AB is a diameter of a circle. BP is a tangent to the circle at B . A straight line passing through A intersects BP at C and the circle at D . Prove that BC^(2)=ACxxCD .

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 .

D, E and F are the mid-points of the sides AB, BC and CA of the DeltaABC . Prove that DeltaDEF=1/4DeltaABC .

In DeltaABC , the bisector AD of /_BAC intersects BC at D . Prove that the DeltaABC is an isoceles triangle.

In the isosceles triangle ABC, AB=AC, if a circle is drawn with the side AB as diamter then the circle intersects the side BC at the point D. When BD=4 cm , find the the length of CD.