Question

Prove the following Theorem 2.33: Given any line, there exists a
point that does not lie on it.

Answer #1

please find the answer below

given a line AB and a point C not on AB, prove that there is a
Ray AD such that AC is between AB and AD

Prove, that between any rational numbers there exists
an irrational number.

(Euclidean and Non Euclidean geometry)
Consider the following statements:
Given a line l and a point P not on the line:
There exists at least one line through P which is
perpendicular to l.
There exists at most one line through P which is
perpendicular to l.
There exists exactly one line through P which is
perpendicular to l.
Prove each statement or give a
counter-example E2 (Euclidean Plane), H2 ,
(Hyperbolic Plane)and the sphere S2 (Spherical
Plane)
( Consider...

1.- Prove the following:
a.- Apply the definition of convergent sequence, Ratio Test or
Squeeze Theorem to prove that a given sequence converges.
b.- Use the Divergence Criterion for Sub-sequences to prove that
a given sequence does not converge.
Subject: Real Analysis

Let S = {0，1} and A be any set. Prove that there exists a
bijection between P(A) and the set of functions between A and
S.

Use De Moivre’s theorem to prove that for any p,q ∈ ?,
that (????? + ? ?????)(????? + ? ?????) = cos(? + ?)? + ????(? +
?)?
Hence simplify (cos 3? +? ???3?) / (cos 5?−? sin 5?)

Prove the Cayley-Hamilton theorem for any n x n square
matrix.

33. Prove that the circumcenter, O, the centroid, G, and
orthocenter, H, lie on a common line, known as the Euler line of
the triangle. (Hint:
One way to approach this proof is to construct the line containing
O and G. Then find a point X on OG such that G is between O and X,
and 2|OG| = |OX|. Show that this point X is on all three altitudes,
and hence X is the orthocenter G.)

Given the following Axioms of four-point geometry, prove whether
or not Axiom 3 is independent of the other Axioms:
1. There are exactly four points
2. Each pair of points are on exactly one line.
3. Each line is on exactly two points.

Prove Fermat’s Little Theorem using induction: ap ≡ a (mod p)
for any a ∈Z.

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