In a circle the angles in the same segment equal one another.
#
Statement
Reason
1.
Circle exists.
given
2.
Angle exists
within segment .
given
3.
Angle exists
within segment .
given
4.
5.
6.
7.
8.
9.
10.
QED. Therefore in a circle the angles in the same segment equal one another.
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Summary
Definition of a point.
Point exists.
I.D1, Definition of a point
Definition of a straight line => A line exists
Straight line exists.I.D4, Definition of a straight line
Definition of a straight line => Choose an arbitrary point
Choose arbitrary point
on line .I.D4, Definition of a straight line
Definition of a circle
Points
and
lie on a circle with center
,
therefore line
equals line .
I.D15, Definition of a circle
Definition of an equilateral triangle.
Lines ,
, and
are equal, therefore
triangle is equilateral.
I.D20, Definition of an equilateral triangle
Draw a straight line from any point to any point.
Draw a straight line from
point to
point .
I.P1, Draw a straight line
Extend a finite straight line continuously in a straight line.
Under construction.
I.P2, Extend a straight line
Draw a circle with center and radius.
Draw circle with
center and
radius .
I.P3, Draw a circle...
All right angles equal one another.
Under construction.
I.P4, All right are equal.
The Parallel Postulate
Under construction.
I.P5, The Parallel Postulate
Things equal to the same thing are equal to each other (lines).
Line equals
line
(by #) and
line equals
line
(by #), therefore
line equals
line .
I.CN1, Equal to the same thing...
Things equal to the same thing are equal to each other (parallelogram and triangles).
Parallelogram equals
double triangle
(by #) and
double triangle equals
double triangle
(by #), therefore
parallelogram equals
double triangle .
I.CN1, Equal to the same thing...
Things equal to the same thing are equal to each other (double angles).
Angle is double
angle
(by #)
and angle is double
angle
(by #),
therefore
angle equals
angle .
I.CN1, Equal to the same thing...
If equals are added to equals, then the wholes are equal.
Under construction
I.CN2, Equals are added to equals...
If equals are subtracted from equals, then the remainders are equal.
Under construction
I.CN3, Equals are subtracted from equals...
Things which coincide with one another equal one another.
Line
equals line .
I.CN4, Things which coincide
The whole is greater than the part.
Under construction
I.CN5, Equals are subtracted from equals...
Construct an equilateral triangle on a given finite straight line.
Construct an equilateral triangle
on line .
I.1, Construct an equilateral triangle...
Place a straight line equal to a given straight line with one end at a given point.
Place line
at the point
equal to the straight line .
I.2, Place an equal straight line
Cut off from the greater of two given unequal straight lines a straight line equal to the less.
Cut off line
from line
equal to .
I.3, Cut off a line equal to another line.
SAS - If two triangles have two sides equal to two sides respectively,
and have the angles contained by the equal straight lines equal,
then they also have the base equal to the base
the triangle equals the triangle,
and the remaining angles equal the remaining angles respectively,
namely those opposite the equal sides.
Under construction
I.4, SAS
In isosceles triangles the angles at the base equal one another, and, if the equal straight lines are produced further, then the angles under the base equal one another.
Under construction
I.5, In isosceles triangles the angles...
If in a triangle two angles equal one another, then the sides opposite the equal angles also equal one another.
Under construction
I.6, In a triangle, the sides opposite equal angles
Given two straight lines constructed from the ends of a straight line and meeting in a point, there cannot be constructed from the ends of the same straight line, and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each equal to that from the same end.
Under construction
I.7, Two lines constructed from the ends of a line and meeting in a point...
If two triangles have the two sides equal to two sides respectively, and also have the base equal to the base, then they also have the angles equal which are contained by the equal straight lines.
Line
equals line
(by step #)
and line
equals line
(by step #)
and line
equals line
(by step #),
therefore angle
equals angle .
I.8, SSS
In a parallelogram, the diameter bisects the areas.
Diameter
bisects parallelogram
forming triangle ,
therefore, parallelogram
is double triangle .
I.34, Diameter of parallelogram.
Triangles on the same base and in the same parallels equal one another.
Triangle
and triangle
are on the same base
and in the same parallels
and
,
therefore triangle
equals triangle .
I.37, Triangles, same base
Find the center of a circle.
Find the center of
circle .
III.1, Find the center of a circle.
The angle at the center of a circle is double the angle at the circumference when the angles have the same circumference as base.
Angle is at the center and
angle is at the circumference,
and they share the same base ,
therefore angle is double
angle .
III.20, Angle at center is double angle at circumference.
