Problem
GEO-B2-M09-P011 Equal Ratios
#11
★★☆☆☆ Level 2 of 5
In triangle \(A(0,0)\), \(B(1,0)\), \(C(0,1)\), points \(P\in AB\), \(Q\in AC\) satisfy \(AP:PB=AQ:QC=m:n\). Prove that \(PQ\parallel BC\) and \(PQ:BC=m:(m+n)\).
Set \(t=\frac{m}{m+n}\). Then \(P(t,0)\), \(Q(0,t)\).
We have \(P(t,0)\), \(Q(0,t)\). The vector \(\overrightarrow{PQ}=(-t,t)=t(-1,1)\), while \(\overrightarrow{BC}=(-1,1)\). Hence \(PQ\parallel BC\) and \(PQ=t\cdot BC\). Since \(t=\frac{m}{m+n}\), we get \(PQ:BC=m:(m+n)\).
A good problem on parametrising a point on a segment.