MathDB

[asy] size(150); dotfactor=4; draw(circle((0,0),4)); draw(circle((10,-6),3)); pair O,A,P,Q; O = (0,0); A = (10,-6); P = (-.55, -4.12); Q = (10.7, -2.86); dot("

O

", O, NE); dot("

O'

", A, SW); dot("

P

", P, SW); dot("

Q

", Q, NE); draw((2*sqrt(2),2*sqrt(2))--(10 + 3*sqrt(2)/2, -6 + 3*sqrt(2)/2)--cycle); draw((-1.68*sqrt(2),-2.302*sqrt(2))--(10 - 2.6*sqrt(2)/2, -6 - 3.4*sqrt(2)/2)--cycle); draw(P--Q--cycle); //Credit to happiface for the diagram[/asy]
In the adjoining figure, every point of circle

\mathit{O'}

is exterior to circle

\mathit{O}

. Let

\mathit{P}

and

\mathit{Q}

be the points of intersection of an internal common tangent with the two external common tangents. Then the length of

PQ

<span class='latex-bold'>(A) </span>\text{the average of the lengths of the internal and external common tangents}\qquad

<span class='latex-bold'>(B) </span>\text{equal to the length of an external common tangent if and only if circles }\mathit{O}\text{ and }\mathit{O'}\text{ have equal radii}\qquad

<span class='latex-bold'>(C) </span>\text{always equal to the length of an external common tangent}\qquad

<span class='latex-bold'>(D) </span>\text{greater than the length of an external common tangent}\qquad

<span class='latex-bold'>(E) </span>\text{the geometric mean of the lengths of the internal and external common tangents}

Problem Statement