Sec. 3.6 Mid-line of an angle-support 45
3.6.3 Mid-lines................................
Given any angle-support|BACsuch that C∈[A,B , there is a unique line l such that
A∈l and for all P=AbutP∈l,|∠BAP|◦=|∠PAC|◦.
Proof.Existence.
This was already shown in 3.6.1 in the case when|BACis straight, so we may
assume thatA,B,Care non-collinear.
By A 5 (v) and 3.5.2, as 0<|∠BAC|◦<180andso0<^12 |∠BAC|◦<90, there is a
half-line[A,PwithPon the same side ofABasCis, such that|∠BAP|◦=^12 |∠BAC|◦.
Then[A,P⊂IR(|BAC)by 3.5.2, so by A 5 (iii)
|∠BAP|◦+|∠PAC|◦=|∠BAC|◦.