Common Tangent

 

Definition:  If a line is a tangent of two circles at the same time,  then this line is known as “Common Tangent” of the two circle.

 

 

 

 

 

 

Property ( External Common Tangent )

                If the circles are on one side of the tangent line only, then this line

                is known as External Common Tangent.

               

                                                                            

 

                If the bigger circle is with radius r₁ and smaller one is with radius r₂ , then

                      

                                       

 

 

        Proof:

                         Recall that radius is perpendicular to the tangent line.

                         Thus, we have a diagram shown on the right.

                         From Pythagorean theorem, the equality holds.

 

 

                       Please notice that the geometry construction for common tangent

                       is derived from this property by calculating the lengths of three

                       quantities in that equalities first.

 

 

 

 

 

 

 

 

Property ( Internal Common Tangent )

                

                If  circles are on different sides on the tangent line, then this line

            is known as Internal Common Tangent.

                   

               If circle P is with radius r₁ and circle Q is with radius r₂ , then

                         

                              

 

            Proof:

 

                           Similarly, use the fact that radius is perpendicular to the tangent.

                           With the diagram shown above and Pythagorean theorem,

                            the equality holds.

 

 

                             Think about how to use ruler and compass to construct the internal common tangent.