Euclid's Postulate 1  A straight line
can be drawn from any point to any point. 
This is the first of Euclid's five geometric axioms. Together,
they form the basis of all Euclidean proofs.

Euclid's Proposition 1  On a given segment, an equilateral
triangle can be constructed. 
Many of Euclid's propositions are constructions. This means that
Euclid proved than certain things can be constructed using a compass
and a straight edge.

Euclid's Proposition 2  Given a line
segment and an end point, a segment of the same length can be constructed. 
This proposition shows that a line segment of a certain length can be constructed
with any point as an end point.

Euclid's Proposition 3  To cut off
from the greater of two given unequal straight lines a straight line equal to the
less. 
This proposition shows that a line segment of a certain length can be constructed
on any larger line segment.

Euclid's Proposition 4  If two sides
of two triangles are equal and the contained angle is equal, the two triangles
are equal. 
This proposition is abbreviated as SAS, short for sideangleside.

Euclid's Proposition 6  If in a triangle two angles equal
one another, then the sides opposite the equal angles also equal
one another. 
This proposition builds the basis of many other properties of
triangles.

Euclid's Proposition 47  In rightangled
triangles the square on the side opposite the right angle equals the
sum of the squares on the sides containing the right angle (Pythagorean
Theorem). 
This proposition is better know as the Pythagorean Theorem. This particular
proposition and its derivatives have perhaps, over the last 23 centuries,
generated more math than any other.

Proof  A trapezoid is isosceles if
and only if the two base angles are equal. 
An isosceles trapezoid is a trapezoid where the nonparallel sides are
equal in length.

The Dykstra Extension to the Pythagorean
Theorem. 
ykstra Extension to the Pythagorean Theorem proves the general
case sgn(alpha + beta  gamma)=sgn(a^2+b^2c^2).

Proof of Right Triangle
Midpoint Vertex Ratio Theorem. 
Proves the conjecture that the ratio of the size of a line drawn from
the midpoint of the hypotenuse of a right triangle to the vertex opposite the
hypotenuse is 1:2.

The Line Connecting the
Midpoints of a Saccheri Quadralateral is Perpendicular to Both of
the Lines. 
The Saccheri Quadralateral was created in an attempt to prove
Euclid's fifth postulate by contradiction. While it did not fulfill
its original purpose, the Saccheri Quadralteral has become an
important part of Hyperbolic Geometry.

The Summit Angles of a
Saccheri Quadralateral are Congruent 
The Saccheri Quadralateral was created in an attempt to prove
Euclid's fifth postulate by contradiction. While it did not fulfill
its original purpose, the Saccheri Quadralteral has become an
important part of Hyperbolic Geometry.
