Geometry with Trigonometry

182 Complex coordinates; sensed angles; angles between lines Ch. 10

10.6 LetA,B,Cbe non-collinear points and takeD∈BC,E∈CA,F∈ABsuch that

BD

BC

=r,

CE

CA

=s,

AF

AB

=t.

Letl,m,nbe, respectively, the lines throughD,E,Fwhich are perpendicular

to the side-linesBC,CA,AB. Show thatl,m,nare concurrent if and only if

( 1 − 2 r)|B,C|^2 +( 1 − 2 s)|C,A|^2 +( 1 − 2 t)|A,B|^2 = 0.

10.7 IfR(Z 0 )is the set of all rotations about the pointZ 0 , show that

(R(Z 0 ),◦)is a commutative group.

10.8 Show that the composition of axial symmetries in two parallel lines is equal

to a translation, and conversely that each translation can be expressed in this

form.

10.9 Prove thatsφ;Z 0 ◦sθ;Z 0 =r 2 (φ−θ);Z 0.

10.10 Prove thatsφ;Z 0 ◦rθ;Z 0 =sφ− (^12) θ;Z 0 .Deduce that any rotation about the pointZ 0
can be expressed as the composition of two axial symmetries in lines which
pass throughZ 0.
10.11 LetF 1 ∼F 2 if the frame of referenceF 2 is positively oriented with respect
toF 1. Show that∼is an equivalence relation.
10.12 Prove the Stewart identity
(z 4 −z 1 )^2 (z 3 −z 2 )+(z 4 −z 2 )^2 (z 1 −z 3 )+(z 4 −z 3 )^2 (z 2 −z 1 )
=−(z 3 −z 2 )(z 1 −z 3 )(z 2 −z 1 ).
Interpret this trigonometrically.
10.13 ProveDeMoivre’s theoremthat
(cosα+ısinα)n=cos(nα)+ısin(nα),
for all positive integersnand all anglesα∈A∗(F),whereıis the complex
number satisfyingı^2 =−1.
10.14 Suppose thatl,m,nare distinct parallel lines. LetZ 1 ,Z 2 ,Z 3 ,Z 4 ∈lwithZ 1 =
Z 2 ,Z 3 =Z 4. Suppose thatZ 5 ,Z 6 ∈n,Z 1 Z 5 ,Z 2 Z 5 meetmatZ 7 ,Z 8 , respectively,
andZ 3 Z 6 ,Z 4 Z 6 meetmatZ 9 ,Z 10 , respectively. Prove that then
Z 9 Z 10
Z 7 Z 8

=

Z 3 Z 4

Z 1 Z 2

.