Question 1088961
<pre><font size = 4><b>

The drawing below has NORTH to the right and SOUTH to the left.

{{{drawing(400,2400/13,-8,5,-1,5,

triangle(3,4,2,0,-7,0), triangle(3,4,2,0,-4,0),

locate(-7,0,Q),locate(-4,0,R),locate(2,0,S), 
locate(3.2,4.4,P),
line(2,0,2.5,0),line(2.8,0,3.3,0),line(3.6,0,4.1,0),

red(arc(-7,0,3,-3,0,22),arc(-4,0,3,-3,0,30),arc(2,0,3,-3,0,76),
locate(-6,.5,x),locate(-3,.6,y),locate(2.6,.8,z))

 )}}}{{{matrix(6,1,

"Given:", QS=a,RS=b,"<SQP"=x,"<SRP"=y,PS=Tower)}}}

We look at the triangles separately
Triangle PQS:

{{{drawing(400,2400/13,-8,5,-1,5,

triangle(3,4,2,0,-7,0), 

locate(-7,0,Q),locate(2,0,S), 
locate(3.2,4.4,P),
line(2,0,2.5,0),line(2.8,0,3.3,0),line(3.6,0,4.1,0),

red(arc(-7,0,3,-3,0,22),arc(2,0,3,-3,0,76),
locate(-6,.5,x),locate(2.6,.8,z)),locate(-2.5,0,a),
red(arc(3,4,3,-3,202,256),locate(1.8,3.6,z-x))

 )}}}{{{matrix(3,1,"<P"+x=z,so,"<P"=z-x))}}}     {{{matrix(3,1,
matrix(1,3,law,of,sines),
PS/sin(x)=a/sin(z-x),
PS=(a*sin(x))/sin(z-x))}}}


Similarly for triangle PRS

{{{drawing(400,2400/13,-8,5,-1,5,

 triangle(3,4,2,0,-4,0),

locate(-4,0,R),locate(2,0,S), 
locate(3.2,4.4,P),
line(2,0,2.5,0),line(2.8,0,3.3,0),line(3.6,0,4.1,0),

red(arc(-4,0,3,-3,0,30),arc(2,0,3,-3,0,76),locate(-3,.6,y),locate(2.6,.8,z),
arc(3,4,3,-3,212,256),locate(1.85,3.6,z-y)

)

 )}}} {{{matrix(3,1,
matrix(1,3,law,of,sines),
PS/sin(y)=b/sin(z-y),
PS=(b*sin(y))/sin(z-y))}}}

We set the two expressions for PS equal to each other:

{{{(a*sin(x))/sin(z-x)}}}{{{""=""}}}{{{(b*sin(y))/sin(z-y)}}}

{{{(a^""*sin(x))*sin(z-y)}}}{{{""=""}}}{{{(b^""*sin(y))*sin(z-x)}}}

{{{(a^""*sin(x))*(sin(z)cos(y)-cos(z)sin(y)^""))}}}{{{""=""}}}{{{(b^""*sin(y))*(sin(z)cos(x)-cos(z)sin(x)^"")}}}

{{{a*sin(x)sin(z)cos(y)-a*sin(x)cos(z)sin(y)}}}{{{""=""}}}{{{b*sin(y)sin(z)cos(x)-b*sin(y)cos(z)sin(x)}}}

{{{b*sin(y)cos(z)sin(x)-a*sin(x)cos(z)sin(y)}}}{{{""=""}}}{{{b*sin(y)sin(z)cos(x)-a*sin(x)sin(z)cos(y)}}}

{{{sin(y)cos(z)sin(x)(b-a)}}}{{{""=""}}}{{{b*sin(y)sin(z)cos(x)-a*sin(x)sin(z)cos(y)}}}

Divide both sides by sin(x)sin(y)sin(z)

{{{(sin(y)cos(z)sin(x)(b-a))/(sin(x)sin(y)sin(z))}}}{{{""=""}}}{{{(b*sin(y)sin(z)cos(x))/(sin(x)sin(y)sin(z))-(a*sin(x)sin(z)cos(y))/(sin(x)sin(y)sin(z))}}}

{{{(cross(sin(y))cos(z)cross(sin(x))(b-a))/(cross(sin(x))cross(sin(y))sin(z))}}}{{{""=""}}}{{{(b*cross(sin(y))cross(sin(z))cos(x))/(sin(x)cross(sin(y))cross(sin(z)))-(a*cross(sin(x))cross(sin(z))cos(y))/(cross(sin(x))sin(y)cross(sin(z)))}}}

{{{(b-a)*(cos(z)/sin(z))}}}{{{""=""}}}{{{b(cos(x)/sin(x))-a(cos(y)/sin(y))}}}

{{{(b-a)*cot(z)}}}{{{""=""}}}{{{b*cot(x)-a*cot(y)}}}

Divide both sides by b-a

{{{cot(z)}}}{{{""=""}}}{{{(b*cot(x)-a*cot(y))/(b-a)}}}

Edwin</pre></font></b>