# Download PDF by Sharipov R.A.: Foundations of geometry for university students and By Sharipov R.A.

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The identify of the ebook is a misnomer. This publication infrequently offers with geometry, it is vitally a bunch idea ebook. while you're getting ready for the foreign arithmetic Olympiad (IMO) and wish to profit geometry, this isn't the booklet to check it from. something yet this booklet. it is a quantity theroy booklet i will say.

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Extra info for Foundations of geometry for university students and high-school students

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3) admits the existence of exactly two points L on the line KM satisfying this condition. The first of them is the above point L. The ˜ second one is the point L that lies on the ray coming out from the point M in the direction opposite to the ray [M K . 3). L means that both points L and L Under this assumption we apply the axiom A13 to the ray [KL ˜ on this ray such that [K M ˜] ∼ and mark a point M = [AC]. Now ∼ ∼ ˜ ˜ from [K L] = [AB] and [K M ] = [AC], applying the item (1) of ˜M ˜] ∼ the axiom A15, we derive [L = [BC].

Then both points B and C lie between the points A and D. Proof. From (A ◮ B ◭ C) it follows that the point A lies on the line BC, while from (B ◮ C ◭ D) it follows that D also lies on the line BC. 1 all of the four points A, B, C, and D lie on one straight line. 2 we find a point E not lying on the line AD (see Fig. 1). Then we apply the axiom A10 to the points C and E. As a result on the line CE we find a point F such that the point E lies in the interior of the segment [CF ]. Let’s draw the lines AE and F B, then consider the triangle F BC.

1 does not change the signs of (k − i) and (q − j). Renumbering the points A1 , . . 6) changes these signs to opposite ones: sign(k − i) → − sign(k − i), sign(q − j) → − sign(q − j). Thus, we see that the equality sign(k −i) = sign(q −j) being valid or not does not depend on a particular choice of the monotonic sequence of points that includes the starting and ending points −−→ −−→ of the vectors AB and CD. 2. The codirectedness is a binary relation in the set of vectors lying on one line. This relation possesses the following properties: −−→ −−→ −−→ (1) AB ⇈ AB for any vector AB; −−→ −−→ −−→ −−→ (2) AB ⇈ CD implies CD ⇈ AB; −−→ −−→ −−→ −−→ −−→ −−→ (3) AB ⇈ CD and CD ⇈ EF imply AB ⇈ EF ; −−→ −−→ −−→ (4) if a vector AB is not codirected with CD, while CD is −−→ −−→ −−→ not codirected with EF , then AB ⇈ EF .