Proposition 7
Given two straight lines constructed on a straight line [from its extremities] and meeting in a point, there cannot be constructed on the same line [from its extremities], and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it.
The strategy for this proof is to perform a proof by contradiction. We will assume that two other straight lines can be constructed in this fashion and will reach a contradiction.
Suppose two straight lines AC and CB are constructed on the straight line AB and meeting at the same point C as shown.
Basically, we have started with a line segment AB and added a point C to complete a triangle. Proposition states that we cannot add another point D on the same side of AB such that AC is equal to AD and BC is equal to BD. Let’s add a point D on the same side of AB as C is.
This will be a proof by contradiction. Assume that AC is equal to AD and BC is equal to BD.
Since AC is equal to AD, the angle ACD is equal to the angle ADC by Proposition 5.
Since angle DCB is smaller than angle ACD (because it is contained in that angle), we know that angle ADC is greater than angle DCB.
The following show that angle CDB is greater than angle DCB:
- Angle CDB is greater than angle CDA (because angle CDA is contained in angle CDB).
- Angle CDA is equal to angle ACD as stated previously.
- Angle ACD is greater than angle DCB (because angle DCB is contained in angle ACD).
[CDB > CDA = ACD > DCB]
Now we shift our attention to triangle BCD. Since BC is equal to BD, Proposition 5 tells us that angle CDB is equal to angle DCB.
But we just showed that if our assumptions were true that angle CDB is greater than angle DCB, which is a contradiction. Therefore, there cannot be constructed on the same line [from its extremities], and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it.
Q.E.D.
That wraps up one week’s worth of these propositions. I hope you are finding it rewarding – George