Two circles c(o, r) and c1(o1, r1) touch each other at p, ex, Mathematics

Two circles C(O, r) and C1(O1, r1) touch each other at P, externally or internally. 

105_circle touch externally or internally.png

Construction: join OP and O1P .
Proof : we know that if two circles touch
each other, the point of contact lies on the line through the centres.
O, P,O1 are collinear
OPO1 is a sttraight line.
Case I: C(o,r) and C1(o1,r1 )touch externally at P.
In ΔOPQ, OP=OQ= r
∠OQP =∠OQP (Angle opposite to equal sides)

Similarly in ΔO1PR, O|P= O|R=r|

∠ORP = ∠RPO -------(2)

Also ∠OPQ = ∠RPO -------(3) (Vertically opposite angles)

From (1),(2) and(3) we get
∠OQP = ∠RPO

But these are alternate angles
=> OP u O1R
Case I I: C(o,r) and C(o1,r1) touch internally at P .
In ΔOPQ, OP=OQ= r

∠OQR = ∠O|RP ...................(4) (angles opposite to equal side)

Similirarly in ΔO|PR, O|P = O|R =r|

 ∠ORP = ∠O1PR ...................(5)

From (4) and (5) we get ∠OQR = ∠O|RP

But these are correspondingg angles.

OQ // O|R

 

 

Posted Date: 9/3/2012 6:02:58 AM | Location : United States







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