I. (a.) Let each of the diagonals AC, BD pass through O.

Since O is the intersection of the diagonals PR and QS of ||gm PQRS,

If PQRS is a rhombus ABCD will be a rectangle. If PQRS is a rectangle ABCD will be a rhombus (b.)—Conversely, if a quadrilateral ABCD be a parallelogram, each of its diagonals will pass through O, the intersection of its medians PR, QS.

If ABCD be a rectangle PQRS will be a rhombus.
If ABCD be a rhombus PQRS will be a rectangle.

[Ex.

II. (a.)-Let one diagl. AC pass through O and cut the other BD at some other point H.

Then (I. b) the figure is not a ||gm. Let AC cut PQ and RS in L and K.

BH=twice EH,

=twice QL,

HD=twice HF,

=twice PL,

.. AC bisects BD;

and... AC bisects all st. lines drawn across the figure parallel to BD.

[Ex. 109.

If PQRS be a rectangle ABAD,

and CB=CD,

and ABCD will have the properties enumerated on p. 102.

(b.)—Conversely, if one diagonal AC of a trapezium bisect the other BD at H, it passes through the intersection O of its medians PR, QS. For as before BH=twice EH,

and HD=twice HF,
EH=HF;

.. AC passes through O.

If AB=AD,

and CB=CD,

PQRS will be a rectangle.

III. (a.)—Let neither of the diagls. AC, BD pass through O, but let their point of intersection at H be on the median QS.

Then ABCD cannot belong either to Class I. (by I. b) or to Class II. (by II. 6).

But as before E and L are mid-points of BH, AH,

and.. EL || AB,

and similarly FK || CD.

But since H is on the diagl. QS of the ||gm PQRS,

EL || FK,

... AB || CD.

[Ex.

.. all lines drawn across the figure parallel to PR are bisected by QS. [Ex.

If PQRS is a rhombus, ABCD will have axial symmetry about QS, and.. will have all the properties enumerated on page 103. (b.)-Conversely, if a trapezium ABCD have AB parallel to CD, then the intersection H of the diagonals AC, BD will lie on the median QS.

For as before EL || AB,

and FK || CD,

... EL || FK;

... H lies on QS;

and if ABCD has axial symmetry about QS,

PQRS will be a rhombus.

[Ex.

IV.-Let neither of the diagls. AC, BD pass through O, and let H not lie on either of the medians.

Then ABCD cannot belong to Class I. (by I. b), to Class II. (by II. b), or to Class III. (by III. b).

It cannot have all lines drawn parallel to a diagonal or a median bisected. It may still, however, have some important properties connected with the circle, the discussion of which requires a knowledge of the properties of that figure.

Our classification of quadrilaterals in general is therefore as exhaustive as Euclid's of parallelograms. But as we have seen the inconvenience of his set of names for his mutually exclusive classes of figures (see p. 101), it will be well to avoid a similar one in giving names to our own.

The properties of the particular trapezium discussed on p. 102 are those which belong to it solely in virtue of its being symmetrical about a diagonal. These it shares with equilateral quadrilaterals. Let all such figures as are symmetrical about a diagonal be called kites. Again the properties of the particular trapezium discussed on p. 103 are those which belong to it solely on account of its being symmetrical about a median. These it shares with the rectangular quadrilaterals. A name is wanted to include all such figures. Prof. Henrici uses symmetrical trapezium.

We doubt whether common usage justifies us in extending the signification of trapezium to any sort of parallelograms, and though, as is the case of the rhombus, the extension might after a time be allowed, the length of its designation is against its general adoption.

Following the analogy of the word kite, we venture to suggest axe or axe-head for all such figures as have one pair of opposite sides parallel and the other pair equal.

DEF.-A quadrilateral ABCD having one pair of parallel sides is often called a trapezoid.

ON LOCI. (SYLLABUS.)

DEF.-If any and every point on a line, part of a line, or group of lines (straight or curved), satisfies an assigned condition, and no other point does so, then that line, part of a line, or group of lines is called the locus of the point satisfying that condition.

In order that a line or group of lines X may be properly termed the locus of a point satisfying an assigned condition A, it is necessary and sufficient to demonstrate the two following associated Theorems:—

(1.) If a point satisfies A, it is upon X; or, if a point is not upon X, it does not satisfy A.

(2.) If a point is upon X, it satisfies A; or, if a point does not satisfy
A, it is not upon X.

1. The locus of a point at a given distance from a given point
is the circumference of a circle having a radius equal to
the given distance and its centre at the given point.
2. The locus of a point at a given distance from a given
straight line is the pair of straight lines parallel to
the given line, at the given distance from it and on
opposite sides of it.

3. The locus of a point equidistant from two given points is
the straight line that bisects, at right angles, the line
joining the given points.

4. The locus of a point equidistant from two intersecting straight lines is the pair of lines, at right angles to one another, which bisect the angles made by the given lines.

INTERSECTION OF LOCI.

If X is the locus of a point satisfying the condition A, and Y the locus of a point satisfying the condition B; then the intersections of X and Y, and these points only, satisfy both the conditions A and B.

1. There is one and only one point in a plane which is equidistant from three given points not in the same straight line.

2. There are four and only four points in a plane each of which is equidistant from three given straight lines that intersect one another but not in the same point.

ON SOLVING GEOMETRICAL PROBLEMS.

I.-METHOD OF INTERSECTION OF LOCI.

Many problems propose, directly or indirectly, the determination of a point which satisfies two given conditions.

Such a problem is best attacked by the Method of Intersection of Loci. The student should proceed as follows:

Consider one of the given conditions only, and find, if possible, the locus of a point satisfying it.

Next consider the other given condition by itself, and find, if possible, the locus of a point satisfying it.

Any point in which the two loci may happen to intersect satisfies each of the given conditions.

1. If the two loci do not intersect there is no such point, and the solution of the proposed problem is impossible.

2. If the two loci intersect in more than one point there is more than one solution of the proposed problem.

3. If a line forming the whole or part of one locus coincides with a line forming the whole or part of the other, every boint on such a line satisfies both conditions, and the problem is an indeterminate one.

The complete solution of a problem includes an investigation—(1) Of the number of possible solutions, and (2) Of the particular conditions under which the problem may become impossible or indeterminate.

As an instance of the application of the method, take the following problem :

To find a point which shall be at given distances from two given straight

lines.