Problem
GEO-B2-M04-P020 Parallelism After a Spiral Similarity
#20
★★★★☆ Level 4 of 5
For a point \(P\), it is known that \(\triangle PAC \sim \triangle PBD\). Additionally, \(AC \parallel PB\). Prove that \(BD \parallel PA\).
From similarity, get the equality \(\angle CAP=\angle DBP\).
From the similarity \(\triangle PAC \sim \triangle PBD\), we get \(\angle CAP=\angle DBP\). Since \(AC \parallel PB\), angle \(\angle CAP\) equals the angle between \(PB\) and \(PA\). Therefore the angle between \(DB\) and \(PB\) equals the angle between \(PA\) and \(PB\). Hence lines \(BD\) and \(PA\) are parallel.
Watch oriented angles; different drawings may place the points differently.