9. A cuts the sides of a ▲ in six points so that three

of them connect with the opposite vertices concurrently. Show that the remaining three connect concurrently with the opposite vertices.

IO. Is the statement of Ex. I true when the As are all described internally upon the sides of the given ▲?

11. If L is an axis of symmetry to the congruent ▲s ABC and A'B'C', and O is any point on L, A'O, B'O, and C'O intersect the sides BC, CA, and AB collinearly.

253°. Theorem.-Two triangles which have their vertices connecting concurrently have their corresponding sides intersecting collinearly. (Desargue's Theorem.)

Also, since AA', BB', CC' are concurrent at O, they divide the angles A' B', C' so that

The converse of this theorem is readily proved, and will be left as an exercise to the reader.

Ex. A', B', C' are points upon the sides BC, CA, AB respectively of the ▲ABC, and AA', BB', CC' are concurrent in O. Then

1. AB and A'B', BC and B'C', CA and C'A' meet in three points Z, X, Y, which are collinear.

2. The lines AX, BY, CZ form a triangle with vertices A", B", C", such that AA", BB", CC" are concurrent in O.

OF RECTILINEAR FIGURES IN PERSPECTIVE.

254°. Def.—AB and A'B' are two segments and AA' and BB' meet in O.

Then the segments AB and A'B' are said to be in perspective at O, which is called their centre of perspective.

The term perspective is introduced from Optics, because an eye placed at O would see A' coinciding with A and B' with B, and the segment A'B' coinciding with AB.

By an extension of this idea O' is also a centre of perspective of AB and B'A'. O is

A

B

B

then the external centre of perspective and O' is the internal

centre.

Def.—Two rectilinear figures of the same number of sides are in perspective when every two corresponding sides have the same centre of perspective.

Cor. 1. From the preceding definition it follows that two rectilinear figures of the same species are in perspective when the joins of their vertices, in pairs, are concurrent.

Cor. 2. When two triangles are in perspective, their vertices connect concurrently, and their corresponding sides intersect collinearly. (253°) In triangles either of the above conditions is a criterion of the triangles being in perspective.

Def.-The line of collinearity of the intersections of corresponding sides of triangles in perspective is called their axis of perspective; and the point of concurrence of the joins of corresponding vertices is the centre of perspective.

255°. Let AA', BB', CC' be six points which connect concurrently in the order written.

These six points may be connected in four different ways so as to form pairs of triangles having the same centre of perspective, viz.,

ABC, A'B'C'; ABC', A'B'C; AB'C, A'BC'; A’BC, AB'C'. These four pairs of conjugate triangles determine four axes of perspective, which intersect in six points; these points are centres of perspective of the sides of the two triangles taken in pairs, three X, Y, Z being external centres, and three X', Y', Z' being internal centres. (254°) The points, the intersections which determine them, and the segments of which they are centres of perspective are given in the following table :—

And the six points lie on the four lines thus,

XYZ, X'Y'Z, X'YZ, XY'Z'.

EXERCISES

1. The triangle formed by joining the centres of the three excircles of any triangle is in perspective with it.

2. The three chords of contact of the excircles of any triangle form a triangle in perspective with the original. 3. The tangents to the circumcircle of a triangle at the three vertices form a triangle in perspective with the original.

SECTION IV.

OF INVERSION AND INVERSE FIGURES.

256°. Def.-Two points so situated upon a centre-line of a circle that the radius is a geometric mean (169°, Def.) between their distances from the centre are called inverse points with respect to the circle.

Thus P and Q are inverse points if
CP.CQ=CB2=R2,

R being the radius.

The OS is the circle of inversion

S

A

C P B

or the inverting O, and C is the centre of inversion.

Cor. From the definition :

1. An indefinite number of pairs of inverse points may lie on the same centre-line.

2. An indefinite number of circles may have the same two points as inverse points.

3. Both points of a pair of inverse points lie upon the same side of the centre of inversion.

4. Of a pair of inverse points one lies within the circle and

one without.

5. P and Q come together at B; so that any point on the circle of inversion is its own inverse.

6. When P comes to C, Q goes to co; so that the inverse of the centre of inversion is any point at infinity.

257°. Problem.-To find the circle to which two pairs of collinear points may be inverse points.

S

T

QC P

U

P, Q, P', Q' are the four collinear points, of which PQ and P'Q' are respecs' tively to be pairs of inverse points.

Through P, Q and through P', Q' describe

any two circles S, S' to intersect in two points U and V. Let the connector UV cut the axis of the points in C, and let CT be a tangent to circle S'. Then C is the centre and CT the radius of the required circle.

Proof.—

CT2=CP'. CQ'=CU . CV = CQ. CP.

Cor. If the points have the order P, P', Q, Q' the centre C is real and can be found as before, but it then lies within both circles S and S', and no tangent can be drawn to either of these circles; in this case we say that the radius of the circle is imaginary although its centre is real. In the present case P and Q, as also P' and Q', lie upon opposite sides of C, and the rectangles CP. CQ and CP'. CQ' are both negative. But R2 being always positive (163°) cannot be equal to a negative magnitude.

When the points have the order P, P', Q', Q, the circle of inversion is again real.

Hence, in order that the circle of inversion may be real, each pair of points must lie wholly without the other, or one pair must lie between the others.