Conjecture: Intersection of diagonals of cyclic quadrilateral lies on a fixed li

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Conjecture: Intersection of diagonals of cyclic quadrilateral lies on a fixed li

kieusuong 1

N Apr 30, 2025 by Aaronjudgeisgoat

Let (O) be a fixed circle and let P be a point outside the circle.
From P, draw a line that intersects the circle at points A and B.
Now, from P, draw another line that intersects the circle at points N and M so that quadrilateral ANMB is cyclic (i.e., lies on the circle).

Let AM and BN intersect at point G. Let AN and BM intersect at point T.
Let PJ be the tangent from P to circle (O), and let J be the point of tangency.

Claim (Conjecture):
As the quadrilateral ANMB varies (still inscribed in the circle), the points T, G, and J always lie on a straight line.
Moreover, this line TJ is perpendicular to the fixed chord AB.

I believe this might be a new result and would appreciate any insights or proof ideas.
Attached is a diagram for reference.

1 reply