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`.

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.

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]

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.

Two chords AB and CD of a circle are perpendicular to each other. If perpendicular drawn to AD from the point of intesection of those two chords AB and CD is produced to meet BC at the point E prove that the point E is the mid- point of BC.

Two tangents are drawn from an external point A of the circle with centre at O which touches the circle at the points B and C. Prove that AO is the perpendicular bisector of BC.

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 .

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.

Manas has drawn a circle with centre O of which AB is a chord. A tangent is drawn at the point B which intersects extended AO at the point. T. If angle BAT=21^(@), find the value of angle BTA.

ABC is an acute-angled triangle .AP is the diameter of the circumcircle of the triangle ABC, EB and CF are perpendiculars on AC and AB respectiely and they intersect each other at the point other at the point Q. Prove that BPCQ is a parallelogram.

Two radii OA and OB of a circle with centre O are perpendicular to each other. If two tangents are drawn at the point A and B intersect each other at the point T, prove that AB= OT and they bisect each other at a ridght angle.