Circle: Semicircle circles and Cosines law

In the figure below , semicircles with centres at A and B and with radii 2 and 1 respectively,are drawn in the interior of , and sharing bases with , a semicircle with diameter JK . The two smaller semicircles are externally tangent to each other and internally tangent to each other and internally tangent to the largest semicircle . A circle centred at P is drawn externally tangent to the two smaller semicircles and internally tangent to the largest semicircle . What is the radius of the circle centred at P .

Solution

Let C be centre of largest semicircle .

By the law of cosines ,

$ \displaystyle \cos \theta =\frac{{A{{P}^{2}}+A{{C}^{2}}-P{{C}^{2}}}}{{2\ AP.AC}}$ 

$ \displaystyle \ \ \ \ \ \ \ \ =\frac{{{{{(2+r)}}^{2}}+1-{{{(3-r)}}^{2}}}}{{2(2+r)}}$

$ \displaystyle \ \ \ \ \ \ \ \ =\frac{{10r-4}}{{2(2+r)}}$

$ \displaystyle \ \ \ \ \ \ \ \ =\frac{{5r-2}}{{2+r}}\ \ \ \ \ \ \ ...........(1)$

$ \displaystyle \cos \theta =\frac{{A{{P}^{2}}+A{{B}^{2}}-P{{B}^{2}}}}{{2\ AP.AB}}$ 

$ \displaystyle \ \ \ \ \ \ \ \ =\frac{{{{{(2+r)}}^{2}}+{{3}^{2}}-{{{(1+r)}}^{2}}}}{{6(2+r)}}$ 

$ \displaystyle \ \ \ \ \ \ \ \ =\frac{{2r+12}}{{6(2+r)}}$

$ \displaystyle \ \ \ \ \ \ \ \ =\frac{{r+6}}{{3(2+r)}}\ \ \ \ \ \ \ ...........(2)$

$ \displaystyle \text{From}\ (1)\ \text{and}\ (2)\ ,\ \ \ \ \frac{{5r-2}}{{2+r}}=\frac{{r+6}}{{3(2+r)}}$ 

$ \displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 15r-6=r+6$

$ \displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 14r=12$ 

$ \displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \therefore \ \ r=\frac{6}{7}$