# Geometric Proofs

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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 side-angle-side.

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 right-angled 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 non-parallel 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^2-c^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.