Thales' theorem

author: Петя
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Българска версия:

    Thales' teorem

            Let's two lines a and b crosses themselves in point О. Points А snd А₁ lyes on a, and В and В₁ lyes on b; and point О

• either doesn't lye over segmnets АА₁ and ВВ₁
•  or lyes over segments АА₁ and ВВ₁.

• Thales' theorem: Lines АВ and А₁В₁ are parallel then and then only when ОА/ОА₁=ОВ/ОВ₁.

• Theorem 1: If lines АВ and А₁В₁ are parallel, then ОА/ОА₁=ОВ/ОВ₁.

• Theorem 2: If ОА/ОА₁=ОВ/ОВ₁, then lines АВ and А₁В₁ are parallel.

    Consequences from Thales' theorem

• Thales' theorem: ОА/ОА₁=ОВ/ОВ₁’

• Consequence 1: ОА/АА₁=ОВ/ВВ₁.

• Consequence 2: ОА₁/АА₁=ОВ₁/ВВ₁.

Parallel project through line

            Let's are given two lines a and l, and these lines are not parallel.If М is unspecified point from a plane, then we build:

1. unique line m,which passes through М and which is not parallel on l;

2.  М₁ - the point of intersection on the lines m and а.

Then:

• point М₁ is parallel projection on М over the line parallel on l;

• the line m is projection line on М;

• for a given line l , we definite projection direction;

• М₁' builds is parallel project.

       When the angle (which is between projection direction l and а) is right then the project is orthogonall, and point М₁ – orthogonall projection on М over а.

            If segment МN is given and M₁, N₁ are parallel projection on the poitns М and N over line а, then  segment М₁N₁ is called parallel projection on the line MN over the line а.

• Theorem: Let's АВ and CD are two segments from a line g, and this line g isn't equal on projection direcktion l. Then the ratio on the segmentsAB/CD in parallel project stayes the same.