(g) A tangent from a point to a circle is the mean proportional between the sects (from the

point to the circumference) of any secant from the same point to the circle.

(h) A circle can be inscribed in, or circumscribed about, any regular polygon.

(i) If the number of sides of an inscribed or a circumscribed regular polygon is increased indefinitely, the perimeter and the area of the polygon approach the circumference and area of the circle as limits.

V. PROPOSITIONS ABOUT FIGURES FORMED BY Two INTERSECTING LINES.

(a) Any two vertical angles are equal.

(b) Bisectors of vertical angles are in a straight line. (c) A quadrilateral is a parallelogram if its diagonals bisect each other.

VI. CONSTRUCTIONS. The given figure is in one plane,
and the construction is to be done in that plane,
except where it can be equally well done in a
different plane, as in (d) and (e) below. Where
the given figure can be in more than one plane, as in
(a) and part of (c), the construction must be done
in one of those planes. In every case there must
be one plane in which the construction is done.
To construct

(a) The perpendicular bisector of a sect.
(b) The bisector of an angle.

(c) The perpendicular to a line from or at a point.
(d) A triangle, given its sides.

(e) An angle equal to a given angle.

(ƒ) A line through a given point || to a given line. (g) The center of a circle, given an arc.

(h) A circle, given an inscribed angle, and the chord it subtends; especially, the locus of

the vertex of a right angle opposite a given
hypotenuse.

(i) A triangle equivalent to a given polygon.
(3) A square equivalent to (1) the sum of two or
more squares, (2) the difference of two

squares, (3) any number of times a given

square.

(k) A third or a fourth proportional to given sects, or a mean proportional between them.

(1) A regular polygon inscribed in, or circumscribed about, a given circle.

VII. FORMULAS.

(a) For the angles of triangles or other polygons.
(1) An exterior angle of a triangle equals
the sum of the non-adjacent interior
angles. The sum of the angles of a
triangle is a straight angle.

(2) The sum of the interior angles of a poly-
gon of n sides is equal to (n-2)
straight angles; the sum of its exterior
angles is equal to two straight angles.
Each angle of a regular polygon of n
(n − 2)

sides equals

n

straight angles.

Concerning the sides of a right triangle. The square of the hypotenuse of a right triangle equals the sum of the squares on the legs; the square of either leg equals the difference of the squares on the hypotenuse and the other leg.

(c) The altitude of an equilateral triangle of side

a

a, is 3; the altitude on b of a triangle

2

(d) The circumference of a circle is 2 πr, where r is the radius, and π= 3.1416-.

(e) Area. The base is represented by b, by b, and b2 when there are two bases; the altitude, by h; the perimeter, by p; the apothem, by a. (1) Of a rectangle or other parallelogram, is bh.

bh
2

(2) Of a triangle, is ; of an equilateral

triangle, using the notation in part (c),

is √s(sa) (8 − b ) (8 — c).

(4) Of a regular polygon isp. a.

(5) Of a circle is 772, where r is its radius.

C. PLANE GEOMETRY PROPOSITIONS THAT CAN BE USED IN SOLID GEOMETRY ONLY WHEN THE ENTIRE FIGURE CAN BE SHOWN TO LIE IN ONE PLANE

I. CONCERNING PARALLELS AND PERPENDICULARS. (a) There can be but one perpendicular to a given line at a given point. Note that in solid geometry any number of planes can contain the given line, and there can be one perpendicular to the line at the given point in each of these planes. (b) A line perpendicular to one of two parallels is perpendicular to the other also.

(c) Lines perpendicular to the same line are parallel.

(d) Lines perpendicular to intersecting lines must meet each other.

II. CONCERNING CIRCLES.

(a) A line perpendicular to a radius at its end on the circumference is tangent to the circle.

(b) Two circles are tangent if they meet at a point on their center line, or are tangent to the same line at the same point; they intersect twice if they meet at a point not on their center line. III. LOCI. The given figure is in a plane. If, as in (a), (e), (ƒ), and (g), it can be in more than one plane, one of those planes must be used. The locus of points in that plane in each case is : (a) Equidistant from two given points, the perpendicular bisector of the sect joining them.

(b) Equidistant from three non-collinear points, the circumcenter of their triangle.

(c) Equidistant from two intersecting lines, the bisectors of the angles between them; from two

parallel lines, the third parallel midway between them.

(d) Equidistant from three lines meeting in pairs, the incenter and three excenters of their triangle. (e) At a fixed distance from a given point, the circumference of a circle drawn with the point as center, and the distance as radius.

(f) At a fixed distance from a given line, two parallels at that distance from the line.

(g) Of the vertex of a given angle subtended by a given fixed sect, the two arcs of circles having that sect as a chord, and the angle inscribed on the arc of that chord; in particular, for a right angle, a circle on the sect as a diameter.

SECTION II. METHODS OF ATTACK

Synthesis and Analysis. The most usual form for a proof is synthetic, that is, the proof starts from the given conditions, and, step by step, builds on them until the conclusion is reached. The demonstrations of plane geometry, to which the pupil is accustomed, are of this form.

While this is an excellent way in which to state a proof that is already known, it is not adapted to discovering a proof, unless the conclusion is an almost immediate consequence of the conditions.

Usually a proof must be discovered by analysis, that is, by working back from the required conclusion to the conditions upon which the conclusion must depend. The different methods of analyzing a proposition will be taken up in some detail. The one which is most nearly pure analysis is called the analysis method, the others being named from some important feature of the process used.

The most gener

Analysis by the Classification Method. ally useful of all the methods of attack is analysis by the classification method. It considers all propositions as belonging to classes according to their conclusions. For example, in using lines, one class proves sects equal, another proves them unequal, a third proves them proportional, while a fourth proves them parallel, etc.

It is evident then, that each new proposition must be proved by some already known proposition of its own