Question 193358
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</pre><font size=4><b>
{{{drawing(350,350,-4,4,-4,4,circle(0,0,3),triangle(-3,0,0,3,3,0),locate(-3.2,0,A),locate(0,3.3,C),locate(3.2,0,B),green(line(0,0,0,3)),locate(0,-.3,D),green(line(-.2,1.5,.2,1.5)),green(line(-1.5,.2,-1.5,-.2)),green(line(1.5,.2,1.5,-.2)))}}}
Prove: {{{CD^2=(AD)(AB)}}}


We know {{{Diameter=AB}}}, and {{{Radius=(1/2)Diameter=(1/2)AB}}}


We can see the triangle is split into 2 Isosceles Right Triangles.
<font color=blue>Theorem</font>:
In an isosceles right triangle, the sides are in the ratio {{{1:1:sqrt(2)}}}
The equal sides make the right angle.


Therefore, equal sides ---> AD=DB=CD=Radius=(1/2)(AB)


Proving: {{{(CD)^2=(AD)(DB)}}} 
={{{(AD)^2=(AD)(AD)}}} --> {{{(AD)^2=(CD)(DB)}}}
={{{(DB)^2=(DB)(DB)}}} --> {{{(DB)^2=(CD)(AD)}}}
={{{(CD)^2=(CD)(CD)}}} --> {{{highlight((CD)^2=(AD)(DB))}}}; *Note: AD=DB=CD=Radius=(1/2)(AB)


Thank you,
Jojo</font>