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ON THE A POSTERIORI DEMONSTRATION OF THE PORISM 
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The porism (allographic) of the in-and-circumscribed triangle, viz. 
If a triangle be inscribed in a conic and two of the sides envelope conics meeting 
the circumscribed conic in the same four points, then the third side will touch a conic 
meeting the circumscribed conic in the four points. 
The following is Poncelet’s demonstration: 
No. 433. In the particular case of the homographic porism, viz.—that in which 
two of the sides of the triangle pass through fixed points and the remaining side 
envelopes a conic having double contact with the circumscribed conic—it is easy to 
see that the lines joining the angles of the triangle with the two fixed points and 
with the point of contact on the third side, meet in a point; this follows at once 
by the principle of projection from the case in No. 431, viz. the case of a triangle 
inscribed in a circle when two of the sides are parallel to given lines and the third 
side touches a concentric circle. Hence, 
No. 531. If there be a triangle inscribed in a conic, and two of the sides envelope 
fixed curves, and the third side envelopes a certain curve ; the lines joining the angles 
of the triangle with the points of contact meet in a point. 
In fact, attending only to the infinitesimal variation of the position of the triangle, 
the curves enveloped by the first and second sides may be replaced by the points of 
contact on these sides, and the curve enveloped by the third side may be replaced 
by a conic having double contact with the circumscribed conic, and the general case 
thus follows at once from the particular one. 
Nos. 162 and 163. Lemma ( x ). If, on the sides of a triangle ABC, there are taken 
any three points L, M, N in the same line, and the harmonics A', B', C' of these 
points (i.e. the harmonic of each point with respect to the two vertices on the same 
side of the triangle), then the lines AA', BB', CC' will meet in a point; and, moreover, 
if A'L, B'M, G'N are bisected in F, G, H (or, what is the same thing, if FA' 2 — FB. FG, 
GB' 2 = GG.GA, HG' 2 = HA.HB), then will the three points F, G, H lie in a line. 
This is, in fact, the theorem No. 164,—In any complete quadrilateral the middle points 
of the three diagonals lie in a line. 
It is now easy to prove a particular case of the allographic porism, viz. 
No. 531. If there be a triangle inscribed in a circle, such that two of the sides 
envelope circles having a common secant (real or ideal) with the circumscribed circle ; 
then will the third side envelope a circle having the same common secant with the 
circumscribed circle. 
For if the triangle be ABC, and the points of contact of the sides CB, CA with 
the enveloped circles and the point of contact of the side AB with the enveloped 
curve, be A', B', C'; if moreover the points of intersection of the circumscribed circle 
and the two enveloped circles be M, iT, and the common secant MN meet the sides 
1 I have not thought it necessary to give the figures; they can be supplied without difficulty.