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Omni-MATH / Let {cal C}_1 and {cal C}_2 be concentric circles, with {cal C}_2 in the interior of {cal C}_1 . From a point A on {cal C}_1 one draws the tangent AB to {cal C}_2 ( Bin {cal C}_2 )…

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problem

Let ${\cal C}_1$ and ${\cal C}_2$ be concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$ . From a point AA on ${\cal C}_1$ one draws the tangent ABAB to ${\cal C}_2$ ( $B\in {\cal C}_2$ ). Let CC be the second point of intersection of ABAB and ${\cal C}_1$ , and let DD be the midpoint of ABAB . A line passing through AA intersects ${\cal C}_2$ at EE and FF in such a way that the perpendicular bisectors of DEDE and CFCF intersect at a point MM on ABAB . Find, with proof, the ratio AM/MCAM/MC .
Plain-text mathematical notation (without MathML)
Let ${\cal C}_1$ and ${\cal C}_2$ be  concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$ . From a point A on ${\cal C}_1$ one draws the tangent AB to ${\cal C}_2$ ( $B\in {\cal C}_2$ ). Let C be the second point of intersection of AB and ${\cal C}_1$ , and let D be the midpoint of AB . A line passing through A intersects ${\cal C}_2$ at E and F in such a way that the perpendicular  bisectors of DE and CF intersect at a point M on AB . Find, with proof,  the ratio AM/MC .
Original LaTeX notation
Let ${\cal C}_1$ and ${\cal C}_2$ be  concentric circles, with ${\cal C}_2$ in the interior of ${\cal C}_1$ . From a point $A$ on ${\cal C}_1$ one draws the tangent $AB$ to ${\cal C}_2$ ( $B\in {\cal C}_2$ ). Let $C$ be the second point of intersection of $AB$ and ${\cal C}_1$ , and let $D$ be the midpoint of $AB$ . A line passing through $A$ intersects ${\cal C}_2$ at $E$ and $F$ in such a way that the perpendicular  bisectors of $DE$ and $CF$ intersect at a point $M$ on $AB$ . Find, with proof,  the ratio $AM/MC$ .

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Source and history

task_id
a1ab67ed-56af-5e0b-8b21-872f7e7ad6a9
task_key
test--a1ab67ed-56af-5e0b-8b21-872f7e7ad6a9
task_revision_id
1
config / split
/ test

Official source

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