What is the rule of the circle ?
Answers
Let a, b and c be the radii of the three circles. We first draw the lines AA', BB' and CC' perpendicular to line L. B'C, CA' and BA" are parallel to line L.
three tangent circles solution
Let x = B'C, y=A'C and z = BA". Pythagora's theorem applied to the right triangle BCB' gives
x 2 + BB' 2 = BC 2
Note that BB' = b - c and BC = b + c. The above equation may be written as follows
x 2 + (b - c) 2 = (b + c) 2
Expand the above, group like terms and solve for x
x 2 = 4 b c
x = 2 Square Root( b c )
Use Pythagora's theorem to triangle AA'C to obtain
y 2 + (a - c) 2 = (a + c) 2
Expand, group like terms and solve for y
y 2 = 4 a c
y = 2 Square Root( a c )
Use Pythagora's theorem to triangle AA"B to obtain
z 2 + (a - b) 2 = (a + b) 2
Expand, group and solve for z
z 2 = 4 a b
z = 2 Square Root( a b )
We now use the fact that z = x + y to write
Square Root( a b ) = Square Root( b c )+ Square Root( a c )
Solve the above for c
Square Root( a b ) = Square Root( c )Square Root ( b )+ Square Root( c )Square Root( a )
Square Root( c ) = Square Root( a b ) / [ Square Root ( b )+ Square Root( a ) ]
c = ( a b ) / [ Square Root ( b )+ Square Root( a ) ] 2
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Substitute a and b by their values to obtain c
c = 10 * 8 / [ Square Root ( 8 )+ Square Root( 10 ) ] 2
c = 2.2 cm (rounded to 1 decimal place)
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In the figure below, three circles are tangent to each other and to line L. The radius of circle A is equal to 10 cm and the radius of circle B is equal to 8 cm. Find the radius of circle C.
The line from the centre of a circle to the centre of a chord is perpendicular to the chord. ... A perpendicular line from the chord to the centre bisects the chord. The angle at the centre is twice the angle standing on the same chord/arc. Angles on the same arc are equal.