Problem
GEO-B1-M03-P021 A Parallel Line and Two Broken Paths
Tangents to a circle at points \(B\) and \(D\) meet at \(P\). A line through \(P\) intersects the circle at \(A\) and \(C\). Through point \(X\) on segment \(AC\), draw a line parallel to \(BD\). It intersects the broken paths \(ABC\) and \(ADC\) at points \(Y\) and \(Z\). Prove that these points divide the two broken paths in the same ratio, measured from \(A\) to \(C\).
C. Hint 1. First prove that \(\frac{AB}{BC}=\frac{AD}{DC}\).
D. Hint 2. Then consider two cases: the line through \(X\) meets the pair of sides near \(A\), or the pair near \(C\).
E. Full solution.
By the tangent-chord theorem, \(\angle PBA=\angle BCA\), and the angle at \(P\) is common to triangles \(PBA\) and \(PCB\). Hence these triangles are similar, so \(\frac{AB}{BC}=\frac{PB}{PC}\).
Similarly, triangles \(PDA\) and \(PCD\) are similar, giving \(\frac{AD}{DC}=\frac{PD}{PC}\). But tangents from the same point are equal: \(PB=PD\). Therefore \(\frac{AB}{BC}=\frac{AD}{DC}\).
Now suppose the line through \(X\), parallel to \(BD\), meets sides \(AB\) and \(AD\). Then by similarity, \(\frac{AY}{AB}=\frac{AZ}{AD}=\frac{AX}{AC}\), so \(Y\) and \(Z\) divide the first links of the broken paths in the same ratio.
If instead the line meets sides \(BC\) and \(DC\), then similarly \(\frac{CY}{CB}=\frac{CZ}{CD}=\frac{CX}{CA}\), so the remaining parts of both broken paths have the same relative size.
Since the two full broken paths have the same ratio between corresponding links, in both cases the point on one broken path and the point on the other represent the same fraction of the path from \(A\) to \(C\).
A. Source analysis. Main objects: two tangents, a chord through their intersection, and two broken paths. The obvious approach is to use the parallel line immediately, but that is not enough without the hidden ratio \(AB:BC=AD:DC\). The needed transformation is to first obtain this ratio by tangent-based similarity. Number of key ideas: 3.
F. Difficulty justification. This is Level 6: a regional-style problem where similarity appears twice and is then used to divide broken paths.
G. Check. This is not a one-step exercise: it requires the tangent-chord angle, two pairs of similar triangles, and then a separate argument with the parallel line.