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Omni-MATH / AB is a diameter of circle O.X is a point on AB such that AX=3BX. Distinct circles ω₁ and ω₂ are tangent to O at T₁ and T₂ and to AB at X. The lines T₁X and T₂X intersect O again a…

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problem

ABA B is a diameter of circle O.XO . X is a point on ABA B such that AX=3BXA X=3 B X. Distinct circles ω1\omega_{1} and ω2\omega_{2} are tangent to OO at T1T_{1} and T2T_{2} and to ABA B at XX. The lines T1XT_{1} X and T2XT_{2} X intersect OO again at S1S_{1} and S2S_{2}. What is the ratio T1T2S1S2\frac{T_{1} T_{2}}{S_{1} S_{2}}?
Plain-text mathematical notation (without MathML)
AB is a diameter of circle O.X is a point on AB such that AX=3BX. Distinct circles ω₁ and ω₂ are tangent to O at T₁ and T₂ and to AB at X. The lines T₁X and T₂X intersect O again at S₁ and S₂. What is the ratio (T₁T₂)/(S₁S₂)?
Original LaTeX notation
$A B$ is a diameter of circle $O . X$ is a point on $A B$ such that $A X=3 B X$. Distinct circles $\omega_{1}$ and $\omega_{2}$ are tangent to $O$ at $T_{1}$ and $T_{2}$ and to $A B$ at $X$. The lines $T_{1} X$ and $T_{2} X$ intersect $O$ again at $S_{1}$ and $S_{2}$. What is the ratio $\frac{T_{1} T_{2}}{S_{1} S_{2}}$?

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Source and history

task_id
873ca5e3-bc64-5ec4-b4eb-b8dd115532b4
task_key
test--873ca5e3-bc64-5ec4-b4eb-b8dd115532b4
task_revision_id
1
config / split
/ test

Official source

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