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Circles

Let s denote the semi-perimeter of a triangle ABC in which BC = a, CA = b, AB = c. If a circle touches the sides BC, CA, AB at D, E, F, respectively, prove that BD = s – b.

Circles

If  a  hexagon  ABCDEF  circumscribe  a  circle,  prove  that AB + CD + EF = BC + DE + FA.

Circles

If a circle touches the side BC of a triangle ABC at P and extended sides AB and AC at Q and R, respectively, prove that AQ = 1/2 (BC + CA + AB) .

Circles

In Figure, from an external point P, a tangent PT and a line segment PAB is drawn to a circle with centre O. ON is perpendicular on the chord AB. Prove that :

(i) PA . PB  =  PN²  – AN²

(ii) PN²  – AN²  =  OP²  –  OT²

(iii) PA.PB  =  PT²

Circles

Prove that a diameter  AB of a circle bisects all those chords which are parallel to the tangent at the point A.

Circles

Prove that the tangents drawn at the ends of a chord of a circle make equal angles with the chord.

Circles

A chord PQ of a circle is parallel  to  the  tangent drawn at a point R of the circle. Prove that R bisects the  arc  PRQ.

Circles

In  Figure,  common tangents AB and CD to two circles  intersect  at  E. Prove that AB = CD.

Circles

In Figure, AB and CD are common  tangents  to  two circles  of  unequal  radii. Prove that AB = CD.

Circles

Prove that the centre of a circle touching two intersecting lines lies on the angle bisector of the lines.