Chapter Three. Building Geometry Solid. Incidence Axioms. I-1: For every point P and for every point Q not equal to P there exists a unique line l incident with P and Q. I-2: For every line l there exist at least two distinct points that are incident with l .
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Building Geometry Solid
I-1: For every point P and for every point Q not equal to P there exists a unique line l incident with P and Q.
I-2: For every line l there exist at least two distinct points that are incident with l.
I-3: There exist three distinct points with the property that no line is incident with all three of them.
B-1 If A*B*C, then A,B,and C are three distinct points all lying on the same line, and C*B*A.
B-2: Given any two distinct points B and D, there exist points A, C, and E lying on BD such that A * B * D, B * C * D, and B * D * E.
B-3: If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two.
B-4:For every linel and for any three points A, B, and C not lying on l:
Corollary (iii) If A and B are on opposite sides of l and B and C are on the same side of l, then A and C are on opposite sides of l.
P-3.3: Given A*B*C and A*C*D. Then B*C*D and A*B*D.
Corollary: Given A*B*C and B*C*D. Then A*B*D and A*C*D.
P-3.4: Line Separation Property: If C*A*B and l is the line through A, B, and C (B-1), then for every point P lying on l, P lies either on ray or on the opposite ray .
If A, B, C are distinct noncollinear points and l is any line intersecting AB in a point between A and B, then l also intersects either AC or BC. If C does not lie on l, then l does not interesect both AC and BC.
Def:Interior of an angle. Given an angle CAB, define a point D to be in the interior of CAB if D is on the same side of as B and if D is also on the same side of as C.P-3.5: Given A*B*C. Then AC = ABBC and B is the only point common to segments AB and BC.P-3.6: Given A*B*C. Then B is the only point common to rays and , and P-3.7: Given an angle CAB and point D lying on line . Then D is in the interior of CAB iff B*D*C.
P3.8: If D is in the interior of CAB; then: a) so is every other point on ray except A; b) no point on the opposite ray to AD is in the interior of CAB; and c) if C*A*E, then B is in the interior of DAE.
Crossbar Thm: If is between and , then intersects segment BC.
and are not opposite rays and D is
interior to CAB.
(b) If a ray emanates from an interior point of ABC, then it intersects one of the sides, and if it does not pass through a vertex, it intersects only one side.
DE AB, there is a unique point F on a given side of line such that ABC DEF.