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20.1: Solid triangles

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截屏2021-03-02 下午4.21.22.png

We say that the point X lies inside a nondegenerate triangle ABC if the following three condition hold:

  • A and X lie on the same side of the line (BC);
  • B and X lie on the same side of the line (CA);
  • C and X lie on the same side of the line (AB).

The set of all points inside ABC and on its sides [AB], [BC], [CA] will be called solid triangle ABC and denoted by ABC.

Exercise 20.1.1

Show that any solid triangle is convex; that is, for any pair of points X,YABC, then the line segment [XY] lies in ABC.

Hint

Assume the contrary; that is, there is a point W[XY] such that WABC.

Without loss of generality, we may assume that W and A lie on the opposite sides of the line (BC).

It imples that both segments [WX] and [WY] intersect (BC). By Axiom II, W(BC) — a contradiction.

The notations ABC and ABC look similar, they also have close but different meanings, which better not to confuse. Recall that ABC is an ordered triple of distinct points (see page ), while ABC is an infinite set of points.

In particular, ABC=BAC for any triangle ABC. Indeed, any point that belongs to the set ABC also belongs to the set BAC and the other way around. On the other hand, ABCBAC simply because the ordered triple of points (A,B,C) is distinct from the ordered triple (B,A,C).

Note that ABCBAC even if ABCBAC, where congruence of the sets ABC and BAC is understood the following way:

Definition 20.1.1

Two sets S and T in the plane are called congruent (briefly ST) if T=f(S) for some motion f of the plane.

If ABC is not degenerate and

\(\blacktriangle ABC\cong \blacktriangle A'B'C',\)

then after relabeling the vertices of ABC we will have

ABCABC.

Indeed it is sufficient to show that if f is a motion that maps ABC to ABC, then f maps each vertex of ABC to a vertex ABC. The latter follows from the characterization of vertexes of solid triangles given in the following exercise:

Exercise 20.1.1

Let ABC be nondegenerate and XABC. Show that X is a vertex of ABC if and only if there is a line that intersects ABC at the single point X.

Hint

To prove the "only if" part, consider the line passing thru the vertex that is parallel to the opposite side.

To prove the "if" part, use Pasch’s theorem (Theorem 3.4.1).


This page titled 20.1: Solid triangles is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Anton Petrunin via source content that was edited to the style and standards of the LibreTexts platform.

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