Tangents and Diagonals in Cyclic Quadrilateral: What Is This About?
A Mathematical Droodle


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Explanation

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Copyright © 1996-2017 Alexander Bogomolny

Tangents and Diagonals in Cyclic Quadrilateral

The applet suggests the following theorem:

Assume that in a cyclic quadrilateral ABCD the tangents to the circumcircle at A and C intersect in S, while those at B and D intersect in T. Then T is incident to the diagonal AC iff S is incident to the diagonal BD.


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The statement is a rephrase of La Hire's theorem:

If point A lies on the polar of point B, then point B lies on the polar of A.

Indeed, AC and BD serves as the polars of S and T, respectively.

But the statement could be redressed in a different manner altogether. As we know, the symmedian at C in triangle BDC and the symmedian at A in triangle ABD both pass through the point T of intersection of the tangents at B and D. The two symmedians coincide iff the diagonal AC and the tangents at B and D are concurrent, i.e., iff T is incident to AC. Then La Hire's theorem tells us that the symmedians of triangles BDC and ABD coincide iff the symmedians of triangles ABC and ADC coincide.

Poles and Polars

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