A sequel to the first six books of the Elements of Euclid, containing an easy introduction to modern geometry, with numerous examples. By John Casey.

76

A SEQUEL TO EUCLID.

Let ABC, A'B'C' be the two As, having the lines
joining their corresponding vertices meeting in a
point 0: it is required to prove that the three points
X, Y, Z, which are the intersections of corresponding
sides, are collinear.
Dem.-From A, B, C let fall three pairs of IJs on
the sides of the A A'B'C'; and from O let fall three is
p', p", p'" on the sides B'C', C'A', A'B'.
Now we have, from Cor., Prop. 3,
AP p", BQ p' CR p"
AP' p"' BQ' p"'" CR' p'
Hence the product of the ratios,
AP BQ CR
AP' BQ' c- = unity.
Again we have, independent of sign, (IV.)
AZ AP BX BQ CY CR
ZB BQT XC - CR' AY AP'
HIence the product of the three ratios
AZ BX CY
ZB' XC' YA
is equal to the product of the three ratios
AP BQ CR
BQ' CR" AP';
and, therefore, equal to unity. Hence, by Cor., Prop. 4,
the points X, Y, Z are collinear.
Cor.-If two As be such that the points of intersection of corresponding sides are collinear, then the
lines joining corresponding vertices are concurrent.