Advanced High-School Mathematics

SECTION 1.2 Triangle Geometry 21

[DXB], so

GX

XI

ID

DH

HB

BG

=−1.

[AY F], so

GA

AI

IY

Y H

HF

FG

=−1.

[CZE] (etc.)

[ABC] (etc.)

[DEF] (etc)

Step 3. Multiply the above five factorizations of −1, cancelling

out all like terms!

18. This time, let the hexagram be in-
    scribed in a circle, as indicated to
    the right. By producing edges [AC]
    and [FD] to a common point R
    and considering the triangle 4 PQR
    prove Pascal’s theorem, namely
    that points X, Y, and Z are co-
    linear. (Proceed as in the proof
    of Pappus’ theorem: consider the
    transversals [BXF], [AY D], and
    [CZE], multiplying together the
    factorizations of−1 which each pro-
    duces.)


19. A straight line meets the sides [PQ], [QR],[RS],and [SP] of the
    quadrilateralPQRS at the pointsU, V, W,andX, respectively.
    Use Menelaus’ theorem to show that

PU

UQ

×

QV

V R

×

RW

WS

×

SX

XP

= 1.