MathLabs

Problem 5

Let ABCD be a quadrilateral inscribed in a circle omega, and let P be a point on the extension of AC such that PB and PD are tangent to omega. The tangent at C intersects PD at Q and the line AD at R. Let E be the second point of intersection of AQ and omega. Prove that B, E, R are collinear.
Step 2 of 7: Use the two tangent similarities
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△PAD∼△PDC,△PAB∼△PBC⟹ADDC=ABBC.\triangle PAD\sim\triangle PDC,\quad\triangle PAB\sim\triangle PBC\Longrightarrow\frac{AD}{DC}=\frac{AB}{BC}.
Detailed analysis

The tangent-chord angle relations at D and B, together with P,A,C collinear, give the two stated similar-triangle pairs. Since PB=PD are tangent lengths, both ratios equal PA/PB, yielding AD/DC=AB/BC.