MHB Euclidean Geometry - Demonstration Exercise

AI Thread Summary
The discussion focuses on demonstrating properties of points and segments in Euclidean geometry, specifically regarding their positions relative to a line. It addresses how points not on a line are positioned on opposite sides of that line and explores the union of segments formed by distinct points on the same line. Participants emphasize the importance of precise definitions, particularly concerning concepts like "opposite sides" and the determination of planes by lines and external points. Clarifications are sought regarding the notation used for points and segments, ensuring a mutual understanding of geometric relationships. The conversation highlights the foundational nature of these proofs in understanding geometric principles.
Samuel Gomes
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(a) Let be m a line and
[IMG]
the only two semiplans determined by m.

(i) Show that: If
[IMG]
are points that do not belong to
[IMG]
such
[IMG]
, so
[IMG]
and
[IMG]
are in opposite sides of m.

(ii) In the same conditions of the last item, show:
[IMG]
and
[IMG]
.

(iii) Determine the union result
[IMG]
, carefully justifying your answer.

(b) Let be
[IMG]
and
[IMG]
4 distincts points in a line
[IMG]
such
[IMG]
and
[IMG]
. Show
[IMG]
and
[IMG]
.

(c) Let be
[IMG]
distincts points in a line m such
[IMG]
. Under these conditions, show 2 distinct segments such the Union of both segments be equal to
[IMG]
, carefully justifying your answer.

Thanks for the help ^^
 
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