In figure, the common tangents, AB and CD to two circles with centres O and O' intersect at E. Prove that the points O, E, O' are collinear. (Figure attached separately.)

Question

In figure, the common tangents, AB and CD to two circles with centres O and O' intersect at E. Prove that the points O, E, O' are collinear.  (Figure attached separately.)

anonymous · Answer

attachment

campbell_st · Answer

in the circle centre O join OA and OC

prove congruency

in triangles OAE   and OCE
                      OE          common side
                   OA = OC   equal radii
         angle OAC = angle OCE   (both 90 - The tangent to a circle is perpendicular to the         radius at the point of contact.)
       
Triangle OAE is congruent to Triangle OCE by Right Hypotenuse side test. 

Since the triangles are congruent and OE is a common then OE are collinear. 

Smilar proof for the other circle... then since OE are collinear and O'E are collinear then OEO' must be colinear

anonymous · Answer

@campbell
Thanks for the help extended but the above will not prove that all three points are collinear.

It will just show that O and E are collinear and O' and E are collinear. That can happen as shown below. Hence, this will not work.

|dw:1329733547628:dw|

campbell_st · Answer

the points lie ont he same line since angle AEC = angle DEB, vertically opposite... and the line OO' bisects each angle... corresponding angles in  congruent triangles...

anonymous · Answer

Yes, now u r closer to what I hv done....
What I have done is to show that sum of all angles about E is 360 degree.
Then showed that 2(<AEO + <AED + <DEO') = 360 deg
so <AEO + <AED + <DEO = 180 degree
so the three points lie on the same straight line

does that make sense!!

campbell_st · Answer

prove congruency eliminates the need for angle sum... just use vertically opposite and OO' is the bisector of corresponding angles in congruent triangles

anonymous · Answer

Thanks campbell for your help....☺