[alogo] Symmetric hexagons

Every symmetric hexagon h= (ABC...) accepts one in- (c1) and one circum-scribed conic (c2) (*).
a) When h is convex, the conics are ellipses, when not, they are hyperbolas.
b) For every point P on c2, the Poncelet conic c(P) tangent to c1 is symmetric too.
c) Is there a homography mapping the corresponding in/circum-scribed of the regular hexagon to c1 and c2 and additionally the regular hexagon to h? (the answer is no)
d) When are the two conics c1, c2 homothetic w.r. to D ? Is it true that sqrt(3)/2 is the only possible homothety ratio? (the answer is yes)

The second example below shows that the [conic hexagons] have also the in/circum-conics, and that these conics are homothetic. For these hexagons it is also interesting to know when they are symmetric.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]

(*) A simple proof: show that the matrix with rows the expressions {x2, xy, y2, x, y, 1}, where we substitute the coordinates of the 3 points (xi,yi) and their negatives (-xi, -yi), has zero determinant. This gives the existence of the circumconic. The other one is obtained by duality. Another proof is contained in the discussion FrenchTrio.html .
For a further study of special symmetric hexagons look at the file: Symmetric_Hexagons_Special.html .

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