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two circles touch each other externally at P. AB is a common tangent to the circles touching them at A & B. the value of angle APB is
Let A be on a circle woth centre O and B be the point on the circle with O' as centre. And AB be the tangent to both circles touching at A and B.

Let the two circles touch at P.

Let the tangent at P meet AB at N.

Now NA and NT are tangents to the the circle with centre O andtherefore NA= NB. So the triangle NAP is isosceles and angles NAP = NPA = x say.

By similar consideration NB and NT are tangents from N to circle with centre O'. So triangle NBP is isosceles with NB=NC and therefore, angles NBP = NPB = y say.

Therefore in triangle ABC, angles A+B+P = x + y + (x+y) = 180

Or 2(x+y) =180.

x+y = 180/2 = 90.

Therefore,

x+y = angle APB =180/2 =90 degree.


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