Tangential dilemma

Geometry Level pending

Two circles of unit radii are present in a plane with their centres $P$ and $Q$ such that $PQ = 6 \text{ units}$ .

Another circle with radius $R$ such that $R>2$ is externally tangent to the other two circles.

A common tangent to all three circles passes through the mid point of $PQ$ .

Find $R$

(This problem is not original)

The answer is 8.

1 solution

Priyanshu Mishra
Aug 26, 2016

Let there be two smaller circles such that $PQ= 6$ , and M be the midpoint of PQ . Let the bigger circle with center O be such that it touches other 2 circles externally. Let its radius be OA $=$ R .

Notice that $\angle PAM= 90^0$ .

As $AP = 1$ , $PM = 3$ , by pythagoras theorem, we have $AM = \sqrt{8}$ .

Now, denote by $\theta$ the angle APM.

In triangle PAM , $\large\ \cos { \theta } =\frac { 1 }{ 3 }$ .

Observe that this is also an angle of triangle OPM.

SO, $\large\ \cos { \theta } =\frac { 1 }{ 3 } =\frac { 3 }{ R+1 }$ .

Forcing $R = \boxed{8}$ .

Tangential dilemma

The answer is 8.

1 solution

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