Tangents and Diagonals in Cyclic Quadrilateral: What Is This About?
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Copyright © 1996-2012 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
- Poles and Polars
- Brianchon's Theorem
- Complete Quadrilateral
- Harmonic Ratio
- Harmonic Ratio in Complex Domain
- Inversion
- Joachimsthal's Notations
- La Hire's Theorem
- La Hire's Theorem, a Variant
- La Hire's Theorem in Ellipse
- Nobbs' Points, Gergonne Line
- Polar Circle
- Pole and Polar with Respect to a Triangle
- Poles, Polars and Quadrilaterals
- Straight Edge Only Construction of Polar
- Tangents and Diagonals in Cyclic Quadrilateral
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Copyright © 1996-2012 Alexander Bogomolny
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