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Omni-MATH / Three noncollinear points and a line ℓ are given in the plane. Suppose no two of the points lie on a line parallel to ℓ (or ℓ itself). There are exactly n lines perpendicular to ℓ …

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problem

Three noncollinear points and a line ℓ\ell are given in the plane. Suppose no two of the points lie on a line parallel to ℓ\ell (or ℓ\ell itself). There are exactly nn lines perpendicular to ℓ\ell with the following property: the three circles with centers at the given points and tangent to the line all concur at some point. Find all possible values of nn.
Plain-text mathematical notation (without MathML)
Three noncollinear points and a line ℓ are given in the plane. Suppose no two of the points lie on a line parallel to ℓ (or ℓ itself). There are exactly n lines perpendicular to ℓ with the following property: the three circles with centers at the given points and tangent to the line all concur at some point. Find all possible values of n.
Original LaTeX notation
Three noncollinear points and a line $\ell$ are given in the plane. Suppose no two of the points lie on a line parallel to $\ell$ (or $\ell$ itself). There are exactly $n$ lines perpendicular to $\ell$ with the following property: the three circles with centers at the given points and tangent to the line all concur at some point. Find all possible values of $n$.

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Source and history

task_id
805a5924-0019-5ea0-bf02-e038c8ba9991
task_key
test--805a5924-0019-5ea0-bf02-e038c8ba9991
task_revision_id
1
config / split
/ test

Official source

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