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Omni-MATH / Let ABC be a triangle. Find all points P on segment BC satisfying the following property: If X and Y are the intersections of line PA with the common external tangent lines of the …

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problem

Let ABCABC be a triangle. Find all points PP on segment BCBC satisfying the following property: If XX and YY are the intersections of line PAPA with the common external tangent lines of the circumcircles of triangles PABPAB and PACPAC , then (PAXY)2+PB⋅PCAB⋅AC=1.\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.
Plain-text mathematical notation (without MathML)
Let ABC be a triangle.  Find all points P on segment BC satisfying the following property:  If X and Y are the intersections of line PA with the common external tangent lines of the circumcircles of triangles PAB and PAC , then ((PA)/(XY))²+(PB⋅PC)/(AB⋅AC)=1.
Original LaTeX notation
Let $ABC$ be a triangle.  Find all points $P$ on segment $BC$ satisfying the following property:  If $X$ and $Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of triangles $PAB$ and $PAC$ , then \[\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.\]

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Source and history

task_id
21ac345b-9b5b-58bc-8042-baa7e345017d
task_key
test--21ac345b-9b5b-58bc-8042-baa7e345017d
task_revision_id
1
config / split
/ test

Official source

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