Question 1207590
Why do we need all those dollar marks? We aren't getting paid!! LOL!! 
<pre>
In triangle PQR, let X be the intersection of the angle bisector of angle P 
with side QR, and let Y be the foot of the perpendicular from X to side PR.
If PQ = 10, QR = 10, and PR = 12, then compute the length of XY.

{{{drawing(400,400,-1,12,-2,11,

line(2.8,9.6,50/11,0),

line(50/11,0,8.0363636363,2.61818181818),
triangle(0,0,10,0,2.8,9.6),  locate(2.8,10.1,P),

locate(0,0,Q), locate(10,0,R), locate(50/11,0,X), locate(8.2,2.94,Y),
locate(2.6,8.7,theta),locate(3.1,8.75,theta), locate(9,.5,2theta),
locate(3.75,.5,3theta), locate(2.2,0,50/11), locate(7.2,0,60/11),

locate(-.9,4.5,PQ=10), locate(8,4.5,PR=12),

locate(4,-1,QR=10)
 



 )}}}

Triangle PQR is isosceles, so let the base angles at P and R be 2&theta;

Angle bisector PX divides QR=10 into the ratio of PQ:PR = 10:12 = 5:6,
so it's easy to show that QX=50/11 and XR=60/11. [Note that 50/11+60/11=10].

{{{matrix(1,2,Angle,PXQ)}}}{{{""=""}}}{{{3theta}}} because it is an exterior angle of triangle PRX.  

Using the law of sines on triangle PXQ

{{{"50/11"/sin(theta)}}}{{{""=""}}}{{{10/sin(3theta)}}}

Multiply both sides by 11/10

{{{5/sin(theta)}}}{{{""=""}}}{{{11/sin(3theta)}}}

We look up the formula {{{sin(3theta)}}}{{{""=""}}}{{{-4sin^3(theta)+3sin(theta)}}}

{{{5^""/sin^""(theta)}}}{{{""=""}}}{{{11^""/(-4sin^3(theta)+3sin(theta))}}}

Multiply through by sin(&theta;):

{{{5}}}{{{""=""}}}{{{11^""/(-4sin^2(theta)+3)}}}

{{{-20sin^2(theta)+15}}}{{{""=""}}}{{{11}}}
{{{sin^2(theta)}}}{{{""=""}}}{{{1/5}}}
{{{sin(theta)}}}{{{""=""}}}{{{1/sqrt(5)}}} => {{{drawing(100,100,-.5,2.5,-.5,2.5,triangle(0,0,2,0,2,1),locate(1,0,2),locate(2.1,.5,1),locate(-.3,.33,theta),locate(1,1.2,sqrt(5)))}}}

Now look at right triangle XYR

{{{sin(2theta)}}}{{{""=""}}}{{{opp/hyp}}}{{{""=""}}}{{{XY^""/(60/11)}}}{{{""=""}}}{{{11*XY/60}}}

{{{2*sin(theta)*cos(theta)}}}{{{""=""}}}{{{11*XY/60}}}

{{{2*(1/sqrt(5))*(2/sqrt(5))}}}{{{""=""}}}{{{11*XY/60}}}

{{{4/5}}}{{{""=""}}}{{{11*XY/60}}}

{{{55XY=240}}}

{{{XY}}}{{{""=""}}}{{{240/55}}}{{{""=""}}}{{{48/11}}}

Edwin</pre>