Crossbar+Theorem

Having completed the proof of Proposition 2.21 part 1 we began the class working on part 2. Proposition 2.21-2 If point D is in the interior of an angle XYZ then: No point on the opposite ray to ray AD is in the interior of angle XYZ. We begin by choosing a point W such that W*Y*T creating a line we will name m. By Proposition 2.15, all points on ray YW are on the same side of line XY which we will rename l. By the same logic all points on ray YW are on the same side of line YZ which we will rename n. Line m intersects line n at point Y. Line m intersects line l at point Y. By lemma 2.8, points T and W are on opposite sides of n. By the same logic, points W and T are on opposite sides of l. By B-4, points Z and W are on opposite sides of line l. By the same logic, points X and W are on opposite sides of line n. Therefore point W is not on the interior of angle XYZ. Part 3: If point D is in the interior of an angle CAB then: If C*A*E, then point B is in the interior of angle DAE. We began the proof with the goal of showing that points D and B are on the same side of line AE, and that points B and E are on the same side of line AD. This would conclude that point B is indeed in the interior of angle DAE. By Definition 2.19, points D and C are on the same side of line AB. By the same logic, points D and B are on the same side of line AC. By lemma 2.8, points C and E are on opposite sides of line AD. Because line AE is the same line as AC, points D and B are on opposite sides of AE. We have established one of our goal statements as true. We need now to prove that points C and B are on opposite sides of line AD. With this information, we will be able to use axiom B-4 to show that points B and E are on the same side of line AD. We attempt a proof by contradiction:Assuming that points B and E are on opposite sides of line AD, Lemma 2.9 tells us that there exists a unique point P such that line AD passes through P and B*P*E. We know that point P lies on line AD, we do not know if point P is an element of ray AD.

At this point, our classroom discussion stalled. We separated into groups and tried to figure out how we can prove which points are in the interior of which angles. We concluded that more thought was necessary to finish the remaining proof. To be continued…

After some discussion of Definition 2.22, which tells us that Ray AD is between Rays AB and AC if rays AB and AC are not opposite rays and point D is in the interior to angle BAC. We move on to our final theorem.

Theorem 2.23 (The Crossbar Theorem) If ray AD is between rays AB and AC, then ray AD intersects segment BD.

To begin, we choose a point E such that E*A*C.By proposition 2.21 part 3 we know that point B is in the interior of angle DAE. By definition 2.19, points E and B are on the same side of line AD. Lemma 2.6 tells us because E*A*C contains point A and A is also a point on line AD, then points C and E are on opposite sides of line AD. By a corollary to B-4 if points B and E are on the same side of line AD and points C and E are on opposite sides of line AD then points B and C are on opposite sides of line AD. This corollary is proved by assuming that the corollary is not true. So points B and C are on the same side of line AD. Since points B and E are on the same side of line AD, by axiom B-4 points E and C are on the same side of AD. This is a contradiction of our hypothesis that so our corollary is true. By lemma 2.9, because points B and C are on opposite sides of line AD, there exists a unique point G such that line AD passes through point G and B*G*C. We know that G is some where on line AD. By theorem 2.11, the Line Separation Property, we know that if we pick a point F on the opposite ray to AD, that if F*A*D, then any point on the line lies either on the ray AF or the ray AD. By proposition 2.21 part 2, we know that no point on ray AF is on the interior of angle BAC so G cannot be on that ray since B*G*C implies that point G in and element of the interior of angle BAC. By the Line Separation Theorem, point G is an element of ray AD, so ray AD intersects segment BC.

There was a typo on the crossbar theorem I corrected. It said F, but it was supposed to be E.