14
THE NEW ZEALAND GAZETTE.
[No. 1

The locus of points equidistant from two intersecting straight lines
consists of the pair of straight lines which bisect the angles
between the two given lines.

The Circle:—
A straight line, drawn from the centre of the circle to bisect a chord
which is not a diameter, is at right angles to the chord; con-
versely, the perpendicular to a chord from the centre bisects
the chord.

There is one circle, and one only, which passes through three given
points not in a straight line.

In equal circles (or in the same circle), (i) if two arcs subtend equal
angles at the centres, they are equal; (ii) conversely, if two
arcs are equal, they subtend equal angles at the centre.

In equal circles (or in the same circle), (i) if two chords are equal,
they cut off equal arcs; (ii) conversely, if two arcs are equal,
the chords of the arcs are equal.

Equal chords of a circle are equidistant from the centre; and the
converse.

The tangent at any point of a circle and the radius through the point
are perpendicular to one another.

If two circles touch, the point of contact lies on the straight line
through the centres.

The angle which an arc of a circle subtends at the centre is double
that which it subtends at any point on the remaining part of
the circumference.

Angles in the same segment of a circle are equal; and, if the line
joining two points subtends equal angles at two other points on
the same side of it, the four points lie on a circle.

The angle in a semicircle is a right angle; the angle in a segment
greater than a semicircle is less than a right angle; and the
angle in a segment less than a semicircle is greater than a right
angle.

The opposite angles of any quadrilateral inscribed in a circle are
supplementary; and the converse.

If a straight line touch a circle, and from the point of contact a
chord be drawn, the angles which this chord makes with the
tangent are equal to the angles in the alternate segments.

If two chords of a circle intersect either inside or outside the circle,
the rectangle contained by the parts of the one is equal to the
rectangle contained by the parts of the other.

Any proof of a proposition will be accepted which appears
to the examiners to form part of a systematic treatment of the
subject: the order in which the theorems are stated is not
imposed as the sequence of their treatment.

In the proof of theorems and deductions from them, the
use of hypothetical constructions will be permitted. Proofs
which are applicable only to commensurable magnitudes will be
accepted.

SECOND SCHEDULE.

EXAMINATION AND CERTIFICATION OF TEACHERS.

THE regulations relating to the examination and classification of teachers
made by Order in Council dated 13th February, 1912, and subse-
quently amended by several Orders in Council, are hereby revoked, and
the following substituted therefor:—

CERTIFICATES: GENERAL CONDITIONS.

1. Certificates distinguished from the highest to the lowest by the
   letters A, B, and C shall be issued by the Director of Education to teachers
   on production of evidence of academic attainment and professional skill.

2. A certificate of Class D, as provided in the regulations for the exami-
   nation and classification of teachers in force prior to the date of the
   operation of this Order (hereinafter called the former regulations), shall be
   issued to any teacher who under the provisions of the former regula-
   tions has already passed the examination for Class D, or who, having already
   fulfilled the requirements of the examination for Class D, except in respect
   of not more than two of the subjects prescribed, furnishes to the Director,
   by examination, evidence of his attainment in the deficient subjects of at
   least as high a standard as was required for a certificate of Class D
   under the former regulations.