I am studying the properties of circles and came across a standard geometric principle that states two tangents can be drawn from any external point to a given circle. While I understand the principle intuitively, I would like to delve deeper into the geometric proof of this concept. Here is my current understanding and approach:
Concept and Definitions:
- Let circle ( O ) have a radius ( r ), and let ( P ) be a point outside the circle such that the distance from ( P ) to the center ( O ) is greater than ( r ).
Proof Approach:
- Draw line segment ( OP ) where ( O ) is the center of the circle and ( P ) is the external point.
- Find point ( M ) on ( OP ) such that ( $PM = r$ ) (thus ( M ) is outside the circle as ( P ) is outside and ( $PM \neq r$ ) of the circle).
- Draw a circle centered at ( M ) with radius ( PM ). This circle intersects circle ( O ) at two points, say ( A ) and ( B ).
- By definition, segments ( PA ) and ( PB ) are radii of the new circle and equal in length, and by the circle's property, they must also be tangents to circle ( O ) since ( $\angle OAP$ ) and ( $\angle OBP$ ) are right angles (tangent-radius theorem).
Questions for Further Clarification:
- Is my construction correct and sufficiently rigorous to prove the theorem?
- Are there alternative methods or simpler geometric constructions that can also demonstrate this principle effectively?