[1] The sum of the signed lengths (measuring from A) of AG+AH is equal to the perimeter s of ABC.

[2] angle(GIH) = pi-A and the circumcircle of triangle AGH passes through I.

[3] Extend sides AB, AC to the isosceles AED with length of equal sides equal to the

[4] The middle J of GH lies on line MN, joining the middles M, N of AE and AD.

[5] Line GH envelopes a parabola touching sides AB, AC at E and D. The parabola touches also MN at its middle K, which is its

[1] Obvious since BG = BF = BC+CH.

[2] angle(GIH) = angle(GIB)+angle(BIH) = angle( BIF)+angle(BIC)-angle(HIC) = angle(BIF)+angle(BIC)-angle(CIF) = angle(BIC)+(angle(BIF)-angle(CIF)) = 2*angle(BIC) = pi-A (since angle(BIC) = (pi-A)/2).

[3] By [1] GA+AH = s and the same is true for AG+GE.

[4] Follows easily from [3]. Notice that M, N are the touch-points with the sides of the excircle centered at I. It is a well known property that AM = AN = s/2 (see Bisector1.html ).

[5] Follows from the characteristic property of tangents to a parabola: The symmetric of a point I (focus) with respect to the tangents lies on a line (directrix). All the properties referred follow easily from the previous remarks.

Notice the location of the

Bisector1.html

TrianglesCircumscribingParabolas.html

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