euclidean geometry

[david]

hello everyone, i'm having problem figuring out what method the author used to get (1), (2) and (3) . help please.

 

[HallsofIvy]

That picture is hard to read but even after I blow it up I cannot find anything on it labeled "d₁".

 

[daon2]

That picture is hard to read but even after I blow it up I cannot find anything on it labeled "d₁".

I spent a few minutes trying to read/decipher the information given also. \(\displaystyle d_1\) is the length of \(\displaystyle AO_1\), but \(\displaystyle r_1, r_2\) are not given, and there are likely other values/relationships missing as well.

 

[david]

r₁=CO_{1 ,} r₂=BO₂

d₂=AO_​2

 

[daon2]

r₁=CO_{1 ,} r₂=BO₂

d₂=AO_​2

Where could a 1 possibly come from? It looks like it may be using law of cosines combined with other similar triangles, but I cannot determine which are similar aside from the two it states are. Best can do:

\(\displaystyle d_1^2 = (X_1O_1)^2+(AX_1)^2 - (X_1O_1)(AX_1)(2\cos\omega)\)