MathLabs

Problem 4

Let A,BA,B be distinct points on a given circle OO, and let PP be the midpoint of ABAB. Let O1O_1 be the circle tangent to the line ABAB at PP and tangent to OO. Let ℓ\ell, different from ABAB, be the tangent to O1O_1 through AA, and let CC be the intersection point other than AA of ℓ\ell and OO. Let QQ be the midpoint of BCBC, and let O2O_2 be the circle tangent to the line BCBC at QQ and tangent to the line segment ACAC. Prove that O2O_2 is tangent to OO.
Step 2 of 5: Extract the tangent-circle ratio
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△TBP∼△ASP,△PAM∼△SPX,XSXP⋅PTPB=12\triangle TBP\sim\triangle ASP,\qquad \triangle PAM\sim\triangle SPX,\qquad \frac{XS}{XP}\cdot\frac{PT}{PB}=\frac12
Detailed analysis

The tangent and chord angle equalities give △TBP∼△ASP\triangle TBP\sim\triangle ASP. Also, the right angle at XX and the definition of MM give △PAM∼△SPX\triangle PAM\sim\triangle SPX. Combining their corresponding side ratios yields XSXP⋅PTPB=12\frac{XS}{XP}\cdot\frac{PT}{PB}=\frac12.