10: CIRCLE Theorems

Picture

The Circle is a fascinating entity. It has certain very cool properties. The first theorem we take up is that if two points on the circumference is joined to the centre it forms an angle which is twice that of any angle formed by the same two points on the far side of the circumference. The proof is shown in the diagram above. The trick is to join the centre with the third point forming thereby two triangles. And it must be known that any triangle formed out of two points on the circumference and the centre has to be an isosceles triangle since the two sides of the triangle invariably are two radii (plural of radius) of the circle. Thus as shown, one of the triangles have two angles x and x and the other y and y. 

Then by another very important theorem of triangle that any external angle of a triangle is equal to the sum of the two opposite angles in the triangle, we come to the angle at the centre as 2x and 2y. And this proves our theorem since 2x +2y is twice x+y.


what if we change the angle from 45 to something else
The first theorem is extended to show that any two angles formed by two points with a third point on the circumference are equal. 

This theorem shows that a perpendicular drawn from the centre of a circle onto a tangent makes a right angle. 

Picture

This is a very unique theorem somewhat difficult to put into words and we recommend that the student try to memorize the theorem visually. The proof depends on the first theorem as shown. 


length of tangents drawn from an external point to the circle

Picture

The length of the tangents drawn from an external point to a circle are equal. The proof is shown by proving that the two triangles are right angled and are shown to be congruent.