NORMAL FORMS. 101 
128. Cross-ratio of Corresponding Areas. Let (ABC) be 
the invariant triangle, Fig. 18, of a collineation T and let P 
and P, be any pair of corresponding points in the plane, P 
being transformed to P,; from P and P, draw perpendiculars 
to BC, CA, and AB (dotted lines). 
The cross-ratio of the pencil through the vertex C is 
sinACP; , sinACP  Pibi , Pb 
k,=C(ABP,P) — GmiaGia ° HmaGi2 W im ° 72h 
But the perpendiculars from P and P, on the sides of the tri- 
angle A BC are proportional to the areas of the triangles of 
which they are the altitudes. Hence 
i NON eee 
oe) Pia 9) Pa we SPIBCs | APEC 
In like manner we have 
ee SAB Ee ieee. aA PBS 
OO IACA ~ AIGA 2 NIRA (AIPA 
We easily verify that k,k,k, = 1. 
Since P and P, were taken to be a pair of corresponding 
points in the plane, and since, by Theorem 9, Chapter I, 
the cross-ratio of the invariant elements and any pair of cor- 
responding elements in a one-dimensional projective transfor- 
mation is constant for all pairs of corresponding elements, we 
have found the following important theorem : 
THEOREM 20. The cross-ratio of the areas of four triangles 
whose vertices are any pair of corresponding points in the collinea- 
tion T and whose bases are any two sides of the invariant triangle 
of T is constant for all pairs of corresponding points. 
129. Implicit Normal Form of Equations of Type I. The 
equations of a collineation are usually given in the form 
ax+by +e 
a’! «x + b/y + ¢!/ 
a «+b y+e' 
AMO reser cee (1) 
i a’ a+bly el! 
When these equations represent a collineation of type I, they 
can be thrown into a normal form in which the constants in 
the equation are the coordinates of the three invariant points 
and the characteristic cross-ratios along the invariant lines. 
