iN THE FIG, THE COMMON TANGENT, AB AND CD TO TWO CIRCLES WITH CENRES O, AND O' INTERSECT AT E. PROVE THAT THE POINTS O, E, O' ARE COLLINEAR

Asked by Divyaa | 25th Mar, 2017, 02:52: PM

Expert Answer:

begin mathsize 16px style Construction colon space Join space OA space and space OC.
In space increment OAE space and space increment OCE comma space we space have
OA equals OC space left parenthesis radii space of space the space same space circle right parenthesis
OE equals OE space left parenthesis common space side right parenthesis
angle OAE equals angle OCE space left parenthesis each space is space 90 degree right parenthesis
rightwards double arrow increment OAE approximately equal to increment OCE space left parenthesis RHS space congruence space criterion right parenthesis
rightwards double arrow angle AEO equals angle CEO space left parenthesis cpct right parenthesis
Similarly comma space angle BEO apostrophe equals angle DEO apostrophe
angle AEC equals angle DEB
rightwards double arrow 1 half angle AEC equals 1 half angle DEB
rightwards double arrow angle AEO equals angle CEO equals angle BEO apostrophe equals angle DEO apostrophe
Since space these space angles space are space equal space and space are space bisected space by space OE space and space straight O apostrophe straight E comma space
straight O comma space straight E space and space straight O apostrophe space are space collinear.
end style

Answered by Rebecca Fernandes | 25th Mar, 2017, 05:39: PM