Advanced Math Test #Geometry_2

Geometry Level 2

Given a circle with its center (O) and a constant diameter AB. A point M is placed on the circumference of the circle such that M isn't overlapping A or B. C is symmetric to O through A. The line, which is perpendicular to AB at C, meets AM at N. BN meets the circumference of the circle at a second point E. BM meets CN at F.

Is the product of A M ‾ × A N ‾ \overline{AM} \times \overline{AN} constant?

Paradoxical Answer No Yes It depends.

Tin Le by Tin Le , 15, Vietnam

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1 solution

Let the radius of the circle be R R and ∠ B A M \angle BAM be θ \theta . Then the given product is 2 R c o s θ × R s e c θ = 2 R 2 2Rcos\theta\times Rsec\theta=2R^2 = c o n s t a n t =\boxed {constant}

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