Question 697522
<pre>
PA=44,PB=294 and PC=56, what is PD?

{{{drawing(400,400,-50,450,-250,250, locate(-9,9,P),

locate(294cos(pi/6),294sin(pi/6),B),locate(44cos(pi/6)+5,44sin(pi/6),A),
locate(231cos(pi/4),-231sin(pi/4),D),locate(56cos(pi/4)+6,-56sin(pi/6),C),
line(0,0,294cos(pi/6),294sin(pi/6)),circle(197.9976524,-4.941993777,162.1465191),line(231cos(pi/4),-231sin(pi/4),0,0)
)}}} 
Draw BC and AD

{{{drawing(400,400,-50,450,-250,250, locate(-9,9,P),

locate(294cos(pi/6),294sin(pi/6),B),locate(44cos(pi/6)+5,44sin(pi/6),A),
locate(231cos(pi/4),-231sin(pi/4),D),locate(56cos(pi/4)+6,-56sin(pi/6),C),
line(0,0,294cos(pi/6),294sin(pi/6)),circle(197.9976524,-4.941993777,162.1465191),line(231cos(pi/4),-231sin(pi/4),0,0),
green(line(294cos(pi/6),294sin(pi/6),56cos(pi/4),-56sin(pi/4)),
line(231cos(pi/4),-231sin(pi/4),44cos(pi/6),44sin(pi/6)))


)}}} 

Angles B and D are inscribed angles intersepting
the same arc AC so they are equal.
Angle P is in both triangles. Therefore triangles 
PAD and PCB are similar, therefore

{{{PA/(PC)}}} = {{{PD/(PB)}}}

Cross-multiplying:

PA×PB = PC×PD

44×294 = 56×PD

12936 = 56×PD

{{{12936/56}}} = PD

231 = PD

So PD = 231

Edwin</pre>