......(3)

BC. AD: CA. BD: AB. CD

= B'C'. A'D' : C'A'. B'D': A'B' . C'D'........................... COR. 1. If A, B, C, D be a harmonic system of points on a circle; A', B', C', D' are also a harmonic cyclic system.

For if the ratios on the left side of (3) are equal, those on the right are also equal.

COR. 2. Combining 3° of the last Article with the previous corollary, it follows that a harmonic system of cyclic points may be inverted into the vertices of a

square.

EXAMPLES.

1. Any two triangles may be placed such that the vertices of the one may be inverses of those of the other taken in any assigned order.

2. Any four points may be inverted into an orthocentric system. [For the latter quadrilateral has the following angles :A', 90-A', 180+4', 90-A'; hence since BOD=A+A', COA =B+90° – A', and A+C+A'+π+A' = 180°; the centres of inversion are the intersections of two known circles BOD and COA.]

3. Each side of a triangle divided by the perpendicular on it from any origin remains unchanged by inversion.

3a. If the origin is the symmedian point of the one triangle, it is also the symmedian point of the other.

4. If a, ẞ, y denote the perpendiculars from any point on a circle, on the sides of an inscribed triangle, then

By sin A+ya sin B+aß sin C=0.

[For let A'B'C' be any three points on a line L, and 0 the origin ;

after inversion O is on the inverse circle L' and

6. The inverse of a figure with respect to a line is its reflexion with respect to the line, and is equal in every respect to the given

one.

7. The inverses A', B', C'... of the points of intersection A, B, C, D of any two figures are the corresponding points of intersection of the inverse figures; and the lines AA', BB', CC'... are concurrent at the centre of inversion.

7a. If two curves touch at A, their inverses touch at A' the inverse of A.

8. A circle coincides with its inverse when the circle of inversion is orthogonal to it.

9. A variable chord AB of a circle, the inverse C' of a fixed point C on it and the centre O are concyclic.

[Since the points A, B, C, ∞ are collinear; their inverses with respect to the given circle are concyclic; i.e., ABC'O is a cyclic quadrilateral.]

10. From any point P on the circum-circle a line is drawn through the symmedian point K, cutting the sides of the triangle ABC in A', B', C', prove the relation 21/PA'=3/PK.

[Employ the properties of Ex. 4 and Art. 15, Ex. 1 (3).]

122. Theorem. The inverse of the circum-circle of a triangle ABC with respect to the in-circle is the ninepoints-circle of the triangle PQR formed by joining the points of contact.

Let X, Y, Z be the middle points of the sides of PQR. From similar triangles we get

OA.OX=0B. OY OC. OZ=r2;

therefore, etc.

=

Mr. Piers C. Ward has applied this property in the following elegant proof of Mannheim's Theorem :

Inverting with respect to the in-circle, the circumcircle inverts into X YZ, that is, a circle passing through

a fixed point Z and of constant radius (r). It therefore envelopes a circle concentric with Z whose radius is equal to the diameter of XYZ; therefore, etc., by Art. 121, Ex. 7a.

EXAMPLES.

1. A variable triangle ABC is inscribed to one and escribed to another circle; prove that the mean centre of the points of contact P, Q, R is a fixed point.

seen.

[This particular case of Weill's Theorem (Art. 53, Ex. 12) is easily For the mean centre of P, Q, R is the point of trisection of the line joining its circum- and nine-points-centres, both of which are fixed; therefore, etc.]

2. If a quadrilateral ABCD be inscribed to one circle and circumscribed to another; prove that the mean centre of its points of contact P, Q, R, S with the inner circle is a fixed point.

[Let W, X, Y, Z be the middle points of the sides of the cyclic quadrilateral P, Q, R, S. Then W, X, Y, Z is a cyclic parallelogram, and is therefore a rectangle. The mean centre of P, Q, R, S is evidently that of the system W, X, Y, Z, or the centre of the circle inverse to ABCD with respect to the other given circle.]

3. The four nine-points-circles of the four triangles formed by taking the vertices of a cyclic quadrilateral in threes pass through a point.

[For the nine-points-circles invert into the circum-circles of the triangles formed by drawing tangents to the circle at the vertices of the quadrilateral; therefore, etc. The more general property for any quadrilateral has been independently demonstrated. Art. 79, Ex. 15.]

SECTION II.

ANGLES OF INTERSECTION OF FIGURES AND OF THEIR INVERSES.

123. The general relations existing between the centres and radii of a circle, its inverse, and the circle of inversion are as follows:

:

Let C, C', O be the centres of the three circles; AB, A'B', MN the extremities of their common diameter; SS' and TT" the direct common tangents intersecting in 0. Join ST and S'T'.

Since AB and A'B' are inverse segments with respect to the circle of inversion, the three circles are coaxal. (Art. 114, Ex. 9.)

Let I and I′ denote the points of intersection of ST and S'T" with the line of centres; by comparing equal triangles OIS and OIT, etc., it follows that ST and S'T' are both perpendicular to AB. The quadrilateral CSS'I' is therefore cyclic; hence the inverse of C is I'; and similarly the inverse of C' is I with respect to the circle of inversion, and therefore :

The centre C of any circle inverts into the inverse I' of the centre of inversion O with respect to the inverse circle C'; and

The inverse I of the centre of inversion O with respect to any circle C inverts into the centre C of the circle inverse to the circle C.*

In the particular case when the inverse circle is a line, the inverse of the centre of a given circle is the reflexion of the origin with respect to the line.

The inverse of ST is the circle on OC as diameter. Again, by similar triangles OC/OC′ = OS/OS′ = CS/C ́S'. d/d' =t/t'=r/r'................

or, say

To find d', t, and r', we have

d'/d=tt'/t2= R2/(d2 ~ p2),

where R is the radius of inversion.

.(1)

a relation which gives the position of the centre C′ of the inverse circle.

*Townsend, Modern Geometry of the Point, Line, and Circle, 1863, p. 373.