Download PLANE GEOMETRY 2 (In the line of Elements, ignoring definitions

PLANE GEOMETRY 2
(In the line of Elements, ignoring definitions and theorems that lack modern rigour and adding
things that are needed for completion. Book 3)
Definitions
Circle,
Axiom 4
: There exists a unique circle with centre as any given point and radius as any given line
Theorem 17
: The bisector of any chord that passes via the centre is perpendicular to the chord, if
the chord does not pass through the centre of the circle.
Lemma 5
Lemma 6
: Two circles that cut or touch do not have same centre.
: Of all the lines that are drawn from a non-centre point of a diameter to the points on the circle,
the one through centre is the largest and the remaining part on the diameter is the least. This fact
is true even if the point is chosen outside of the circle.
: If more than two lines can be drawn from an interior point of a circle to the points on
the circle, then that interior point is the centre.
Lemma 7
Theorem 18
: Two circles cannot meet at more than two points.
Lemma 8
: If two circles touch, their centres and point of contact are collinear.
Theorem 19
: Equal chords are at equidistant from the centre.
Construction 7 : Construct a tangent to a given circle from a given point.
Theorem 20
: The line joining centre and point of contact is perpendicular to the tangent.
Theorem 22
Theorem 22
Theorem 23
: Angle at the centre is twice the angle on the circumference.
: Angles in the same segment are equal.
: Opposite angles in a cyclic quadrilateral are two right angles.
Construction 8 : Construct the circle whose segment is given.
Construction 9 : Bisect a given arc.
Theorem 24
: Angle in a semi-circle is right. Angle in major segment is acute and that in minor
segement is obtuse.
Theorem 25
: Alternate segment theorem.
Construction 10: Construct a segment of a circle on a given line and of a given angle.
Construction 11: Cut off a segement of given angle from a given circle.
Theorem 26
: Secant theorem