Geometry with Trigonometry

40 Distance; degree-measure of an angle Ch. 3

In turn

s=

r

1 −r

and so s−sr=r,

giving
s=r( 1 +s) and thus r=

s

1 +s

3.4 Thecross-bartheorem.........................

3.4.1

The cross-bar theorem.Let A,B,C be non-collinear points, X=A any point on
[A,B and Y=A any point on[A,C.IfD=A is any point in the interior region
IR(|BAC),then[A,D∩[X,Y]=0./

Proof.IfDis on[A,B or
[A,Cthe result is clear, so
we turn to other cases. By
3.1.2 there is a pointE=
A such that A∈[E,X].
ThusXandEare on dif-
ferent sides of the lineAC.

A

B

X

Y

C

D

E

Figure 3.5. The Cross-Bar Theorem.

Then by 2.2.3(iv) every point of[Y,E (other thanY) is on one side ofAC, while
every point of[A,D(other thanA) is on a different side ofAC; thus[A,D does
not meet[Y,E]. Moreover the other points of the lineADare on one side of the
lineAB, while the points of[E,Y (other thanE) are on the other side ofAB.On
combining these two, we see that the lineADdoes not meet the side[E,Y]of the
triangle[E,X,Y].AsADdoes meet the side[E,X]of that triangle, we see by 2.4.3
thatADmust meet the third side[X,Y]of that triangle at some pointF.AsF∈
[X,Y]⊂IR(|BAC),Fmust be on the part ofADinIR(|BAC),thatisF∈[A,D.

3.5 Degree-measure of angles

3.5.1 Axiom for degree-measure

Primitive Term. There is a function||◦on the set of all wedge-angles and straight-
angles, intoR. Thus with each angleα, either a wedge-angleα=∠BACor a straight-
angle with support|BAC, there is associated a unique real number|α|◦, called its
degree-measure.