CIRCLE

SOME RESULTS ON CIRCLES

THEOREM 1.

IF TWO ARCS OF A CIRCLE ARE CONGRUENT THEN THE CORRESPONDING CHORDS ARE EQUAL.

THEOREM 2.

THE PERPENDICULAR FROM THE CENTRE OF A CIRCLE TO A CHORD BISECT THE CHORD.

THEOREM 3. 

THE LINE JOINING THE CENTRE TO THE MID POINT OF A CHORD IS PERPENDICULAR TO THE CHORD.

THEOREM 4. 

THE PERPENDICULAR BISECTORS OF THE CHORDS OF A CIRCLE INTERSECT AT ITS CENTRE.

THEOREM 5. 

THERE IS ONE AND ONLY ONE CIRCLE PASSING THROUGH THREE NON COLLINEAR POINTS.

(I) AN INFINITE NUMBER OF CIRCLE CAN BE DRAWN TO PASS THROUGH A SINGLE POINT.
(II) AN INFINITE NUMBER OF CIRCLE CAN BE DRAWN TO PASS THROUGH TWO GIVEN POINTS.
(III) A UNIQUE CIRCLE CAN BE DRAWN TO PASS THROUGH THREE GIVEN NON COLLINEAR POINTS.

THEOREM 6. 

EQUAL CHORDS OF CONGRUENT CIRCLES ARE EQUIDISTANT FROM THE CORRESPONDING CENTRES ARE EQUAL.

THEOREM 7.

CHORDS WHICH ARE EQUIDISTANT FROM THE CORRESPONDING CENTRES ARE EQUAL.

THEOREM 8. 

EQUAL CHORDS OF A CIRCLE ARE EQUIDISTANT FORM THE CENTRE.

i,e., CHORDS AT EQUAL DISTANCE FOR THE CENTRE ARE EQUAL.

THEOREM 9. 

OF ANY TWO CHORDS OF A CIRCLE, THE GREATER CHORD IS NEARER TO THE CENTRE.

THEOREM 10. 

THE ANGLE SUBTENDED BY AN ARC OF A CIRCLE AT THE CENTRE IS DOUBLE THE ANGLE SUBTENDED BY IT AT ANY POINT ON THE CIRCUMFERENCE OF CIRCLE.

THEOREM 11. 

THE ANGLE IN A SEMI CIRCLE IS A RIGHT ANGLE.

THEOREM 12. 

ANGLE IN THE SAME SEGMENT OF THE CIRCLE ARE EQUAL.

THEOREM 13. 

IF TWO CHORDS AB AND CD OF A CIRCLE INTERSECT INSIDE OR OUTSIDE THE CIRCLE WHEN  PRODUCED AT A POINT P.
THEN, AP X PB = DP X PC