PROPOSITION.

(26.) To draw a line passing through a given point, and crossing a given triangle, in such a manner that the sum of the perpendiculars on it from the two vertices on one side of it shall be equal to the perpendicular on it from the other verter placed on the other side of it.

Let D be the given point, and ABC the given triangle, and let DE be the required line, so that A E and B G taken together are equal to CF. Draw C H from C to the middle point H of A B, and draw H K perpendicular to D E.

E

K

H

B

In the trapezium A E G B, the parallels A E, H K, and B G are in arithmetical progression ; A therefore the sum of A E and B G is equal to twice HK; but this sum is also equal to CF. Therefore CF is equal to twice HK. Since CF and HK are parallel, the triangles H L K and CFL are similar, and therefore

CF: HK:: CL:LH.

But CF is equal to twice H K, and therefore CL is equal to twice LH, or LH is one third of CH. Since CH is given in magnitude and position, the point L is given. Hence the problem is solved by drawing a line from any angle C of the triangle, bisecting the opposite side A B, and taking on this one third of it HL. The line drawn from the given point D through the point L will be that which is required.

If the given point happen to be the point L itself, any line whatever passing through it will have the proposed property, and hence we have the following porism: A triangle being given in position, a point may be determined, such that any line being drawn through it, the sum of the perpendiculars from two angles of the triangle placed on one side of it, shall be equal to the perpendicular from the remaining angle and the other side.'

The point L is evidently the centre of gravity of equal masses placed at the three vertices, or, considered mathematically, it is the centre of the mean distances of the three points A B C.

This porism is only a particular case of a much more general one: 'Any number of points being given in the same plane, a point may be found through which any line whatever being drawn, it will pass amongst the points in such a manner, that if perpendiculars be drawn from them upon the line the sum of the perpendiculars at the one side will be equal to the sum of the perpendiculars on the other side.' In this case, as in the former, the sought point is the centre of mean distances. The same porism may receive another modification which generalizes it further. Any number of points being given in the same plane, to determine the condition under which a right line may be drawn amongst them, so that the sum of the perpendiculars from the points on one side shall exceed the sum of the perpendiculars from the points on the other side by a given line."

6

*

* See Algebraic Geometry, p. 34.

In this case, it may be proved that the line must be a tangent to a circle, whose centre is the centre of mean distances, and whose radius is equal to the given line divided by the number of given points.

If the given points be not in the same plane, the porism may be made still more general: Given any number of points in space, to determine a plane passing among them, so that the sum of the perpendiculars from the points on one side shall exceed the sum of the perpendiculars from the points on the other side by a given line.'

In this case the plane much touch a sphere whose centre is the centre of mean distances, and whose radius is the given line divided by the number of points.

If the sum of the perpendiculars on one side be equal to those on the other, the given line and the radius of the sphere vanish, and the sphere is reduced to its centre, i. e. the centre of mean distances. Hence, if a plane be drawn through the centre of mean distances, the sum of the perpendicular from the points on the one side is equal to the sum of the perpendiculars from the points on the other side.'

PROPOSITION.

(27.) A circle and a straight line being given in position, a point may be found such that any right line drawn from it to the given line shall be a mean proportional between the parts of the same line, intercepted between the given right line and the circumference of the given circle.

Let A B be the given right line, H K F the given circle, and D the sought point. Draw GDI perpendicular to AB through D, and also any other line C D F. Also join CI and draw H K.

G

H

E

K

The square of CD is equal to the rectangle CEx CF; but it is also equal to the squares of C G and G D, and the c rectangle C E x CF is equal to the rectangle CK × CI. Hence the rectangle C K x CI is equal to the A sum of the squares of CG and G D. The square of GD is equal to the rectangle G H × GI; therefore the rectangle G H× GI, together with the square of C G, is equal to the rectangle C K × CI. Also the square of CI is equal to the sum of the B squares of CG and G I. But the square of CI is equal to the rectangle CKXCI, together with CIX K I, and the sum of the squares of CG and GI is equal to the square of C G, together with the rectangles GH × GI and GI× HI. Taking away from these equals the rectangle CKC I, and its equivalent the rectangle G H × GI, together with the square of G C the remainders, the rectangles CIX IK and GI× IH are equal. Hence we have

GIIC::IK: IH.

F

Therefore, in the triangles CIG and HIK the angle I is common, and the sides which include it are proportional, and therefore the triangles are similar; but G is a right angle, and therefore HKI is a right angle, and therefore H I is a diameter. Since, then, HI passes through the centre of the given circle, and is perpendicular to AB

the given right line, it is given in position. Also GH and GI are given in magnitude, and therefore G D, which is a mean proportional between them, is given in magnitude, and therefore the point D is given in position.

(28) There is between local theorems and porisms a close analogy. In fact, every local theorem may be converted into a porism; but, on the contrary, every porism cannot be converted into a local theorem. In local propositions the indeterminate is always a point, the position of which is restricted, but not absolutely fixed by the given conditions. Such may always be expressed as a porism. But this class of propositions is more general than geometric loci; the indeterminate may be a line, the direction of which is not restricted by the conditions, but which is otherwise limited, as, for example, to pass through a given point, or to touch a given circle. It may also be a plane similarly restricted to pass through a given point, or to touch a given sphere. Instances of these have been given in (26).

Porisms, in common with geometric loci, take their rise from the conditions of a problem becoming indeterminate. This may happen in two ways. The number of conditions may not be sufficient, or among the given conditions there may exist some particular relation, by which some one or more of them may be deduced from the others. Thus, for the determination of a triangle, three conditions are necessary; and such a problem becomes manifestly indeterminate, if only two conditions be given. But even though three be given, the problem will still be indeterminate, if any one of the three can be inferred from the other two. For example, suppose the base of a triangle, the point where the perpendicular intersects it, and the difference of the squares of the sides be given, the problem to determine the triangle is indeterminate, because the difference of the squares of the sides is equal to the difference of the squares of the segments of the base, and may, therefore, be inferred from the base and the point of section.

The geometrical circumstances by which determinate problems in Geometry are converted into porismatic and local problems, are precisely similar to those under which the solution of an algebraical question becomes indeterminate. In such a question there should be as many equations as unknown quantities, and the problem is indeterminate evidently if there be less. But it may also be indeterminate, even if the number of equations be equal to that of the unknown quantities, and will be so when any one of the equations can be deduced from the others. It may in general be observed, both in geometrical and algebraical problems, that the number of independent conditions should be equal to the number of quantities sought, and should neither be more nor less. If they be more, the results may be inconsistent, and if they be less, the solution will be indeterminate.

SECTION VI.

Invention of Exercises.

I SHALL not attempt here to furnish the student with a collection of particular problems for the exercise of his ingenuity. To accomplish this would swell the bulk of the volume, and increase its price without any adequate advantage. I consider it better to offer such observations as may give a facility in the invention of geometrical exercises, and which, though applied to particular cases, may be very easily generalized.

In problems respecting the construction of a triangle, three independent data are necessary. Of the various lines and angles which have remarkable geometrical connection, or relation with a triangle, any three may be given to construct the triangle; and hence innumerable problems arise, which it would be almost as vain to attempt to enumerate as to solve. We shall, however, mention some of the most remarkable magnitudes thus connected with a triangle, and which most commonly form the data in such problems:

R, the radius of the circumscribed circle.

r, the radius of the inscribed circle.

b', the line from the vertex bisecting the base.

B', the line drawn to the base bisecting the vertical angle.

S, the area.

P, the perimeter.

In general, any three of these may be assumed as data for the construction of the triangle. It is obvious, however, that relations may be instituted between innumerable other magnitudes and the triangle, so as to increase the number of data without limit.

Besides this, the data may be, not any of the single magnitudes above mentioned, but some magnitudes which depend on their combination in pairs by certain geometrical operations. Thus to determine the triangle, we may be given the base, the sum of the sides, and the difference of the base angles. In general, new data may be found by combining the above in pairs, by addition, or subtraction; or, if they be lines, by rectangles, squares, or ratios. By combining the two sides a, c alone, we obtain the following additional data:

axc, the rectangle under the sides.

Without pursuing this subject further, the student will perceive how he may invent for himself numberless problems respecting the construction of triangles.

In addition to the properties of right-angled triangles proved in the Elements, the student will find the following an useful exercise :

Let a, b be the sides, c the hypotenuse, s, s the segments of the hypotenuse, p the perpendicular on the hypotenuse, and let the other letters retain the significations already assigned to them; the square of a line being signified by the number 2 placed above it.

1. (a2 + b2) — (s2 + s12) = 2 p2.

3. r Pcp, (this is a property of every triangle.) 4. given r and c, to find the triangle.

5. given r and a + b.....

6. given r and a..

7. abr P = 2 r (c + r).

8. pc = r P = 2 r (c + r). 9. c (pr) = r2.

10. cp = r2 + cr.

12. (c+p) (a + b)2 = p.

13. {a+b+c+p} × {(c + p) − (a + b) } = p2. 14. The triangle formed by a + b, p and c+p is right-angled. 15. (c – p) – (a - b) = p.

16. {(cp) + (a − b) } × { (c − p) − (a — b) } = p2.

17. The triangle formed by (c - p), (a - b), and p, is right-angled. A numerous class of problems have for their object to draw a right line from a given point intersecting a circle under given conditions. The conditions by which the secant may be restricted in this case are infinitely various. They may have reference to any of the three intercepts between the given point and the two points of intersection with the circle, and any given relation may be required to subsist between any two of these intercepts. Again, in addition to a circle, a right line may be intersected by the line from the given point. This will still further increase the number of relations which may be selected as data for the problem, or, instead of a right line, a second circle may be given.

An extensive class of problems respecting maxima and minima arise out of these considerations; for, in most instances, some of the intercepted parts admit of limiting magnitudes.

Figures inscribed within and circumscribed around each other under given conditions, form also a large class of exercises. In these the considerations of maxima and minima may be introduced with advantage.