Power of a Point (AIME / USAJMO / AMO)

Power of a Point: core idea

No proof submission required — answers only.

• 1. **(Proof track.)** Prove the tangent–secant theorem. Tangent length $t=10$, external part $e=5$; enter the whole secant length.

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• 2. **(Proof track.)** Prove that the radical axis of two circles is a line, and find the length of the common chord of $x^2+y^2=25$ and $(x-3)^2+(y-4)^2=16$. Enter the length.

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• 3. **(Proof track.)** Prove power of a point is well-defined (independent of the line). Enter the power of $(0,0)$ w.r.t. $(x-3)^2+(y-4)^2=4$.

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• 4. **(Proof track.)** Using power, prove a formula for the length of a chord at distance $d$ from the center of a circle of radius $R$. For $R=13$, $d=5$, enter the chord length.

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• 5. **(Proof track.)** Prove that angle in a semicircle is right, then with diameter $10$ and chord $6$ from an endpoint, enter the distance from the center to that chord.

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• 6. **(Proof track.)** Prove Ptolemy’s theorem for a cyclic quadrilateral and apply: cyclic quad with sides $3,5,4,6$ in order and one diagonal $5$ — enter the other diagonal (assume the configuration where Ptolemy applies cleanly with product $39$). If ambiguous, enter $39/5$.

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• 7. **(Proof track.)** Show two tangents from an external point are equal. If each tangent is $12$ and the distance from center to external point is $13$ with radius $5$, enter the length of the contact chord $AB$.

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• 8. **(Proof track.)** Prove the intersecting secants theorem. External segments $3$ and $4$, far points making wholes $x$ and $9$; enter $x$.

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• 9. **(Proof track.)** Using directed power, prove a signed version of the chord theorem. Enter the (unsigned) power of $(3,4)$ w.r.t. $x^2+y^2=9$.

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