Symmetric_hexagons

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.

(*) 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 (x_i,y_i) and their negatives (-x_i, -y_i), 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 .

Symmetric hexagons

See Also