Post by Admin on Oct 13, 2016 9:42:44 GMT
Gary
For the proofs below use diagram 26 on page 306 but with $\angle c\widetilde{p}m$ and $\angle pmc$ not already labelled, since these are the answers to the exercises.
a) In figure 26 on page 306 show that $\angle pmc=\Pi$.
Draw a line parallel to the horizon through $p$ which intersects $L$ in $a$.
From the diagram 26 we can see that $\angle apm+\pi/2=\Pi+\pi/2$ and so $\angle apm=\Pi$
Since $pa$ is parallel to $cm$ by construction it follows that $\angle pmc=\angle apm=\Pi$. Done.
b)Show that it follows from a) above that $\angle c\widetilde{p}m$ equals $(\Pi/2)$.
From $\triangle$ $c\widetilde{p}m$ we can write $\angle c\widetilde{p}m=\pi/2-\angle pcm$
Since $\triangle$ $pcm$ is isosceles it follows that $\angle pcm=(\pi-\Pi)/2$ and so
$\angle c\widetilde{p}m=\pi/2-(\pi-\Pi)/2=\pi/2-\pi/2+\Pi/2=\Pi/2$. Done.
Vasco
For the proofs below use diagram 26 on page 306 but with $\angle c\widetilde{p}m$ and $\angle pmc$ not already labelled, since these are the answers to the exercises.
a) In figure 26 on page 306 show that $\angle pmc=\Pi$.
Draw a line parallel to the horizon through $p$ which intersects $L$ in $a$.
From the diagram 26 we can see that $\angle apm+\pi/2=\Pi+\pi/2$ and so $\angle apm=\Pi$
Since $pa$ is parallel to $cm$ by construction it follows that $\angle pmc=\angle apm=\Pi$. Done.
b)Show that it follows from a) above that $\angle c\widetilde{p}m$ equals $(\Pi/2)$.
From $\triangle$ $c\widetilde{p}m$ we can write $\angle c\widetilde{p}m=\pi/2-\angle pcm$
Since $\triangle$ $pcm$ is isosceles it follows that $\angle pcm=(\pi-\Pi)/2$ and so
$\angle c\widetilde{p}m=\pi/2-(\pi-\Pi)/2=\pi/2-\pi/2+\Pi/2=\Pi/2$. Done.
Vasco