I'm having trouble with proposition 7 in book 1 of the elements.
In the text I'm reading:
"Since AC equals AD, therefore the angle ACD equals the angle ADC. Therefore the angle ADC is greater than the angle DCB. Therefore the angle CDB is much greater than the angle DCB."
I'm trying to follow along, so when I set it up I picked D randomly and it happened to fall inside the the triangle ACB.
This through me off because when D is inside ACB, the line CB no longer divides the angle ACD when considering the angle DCB.
It's obvious DCB is smaller than ADC and ACD when it is divided by this line but is this the only thing that justifies the statement: "Therefore the angle ADC is greater than the angle DCB" or is there something in proposition 5 that makes this more obvious, or am I missing something else entirely? Or is it the case that that when D falls within the triangle, a different proof is needed?
In the text I'm reading:
"Since AC equals AD, therefore the angle ACD equals the angle ADC. Therefore the angle ADC is greater than the angle DCB. Therefore the angle CDB is much greater than the angle DCB."
I'm trying to follow along, so when I set it up I picked D randomly and it happened to fall inside the the triangle ACB.
This through me off because when D is inside ACB, the line CB no longer divides the angle ACD when considering the angle DCB.
It's obvious DCB is smaller than ADC and ACD when it is divided by this line but is this the only thing that justifies the statement: "Therefore the angle ADC is greater than the angle DCB" or is there something in proposition 5 that makes this more obvious, or am I missing something else entirely? Or is it the case that that when D falls within the triangle, a different proof is needed?
