Note (1) Hence the problem-To describe a triangle which shall be inscribed in circle (radius R) and circumscribed about circle (radius r)—is impossible, unless the square on the distance between the centres of the circles R2 2 Rr; and, if that is the case, an infinite number of such triangles can be described: in this latter case the Problem is said to be indeterminate.

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Note (2)-The generalisation of this Theorem for any polygon is called Poncelet's Theorem. A proof of it will be found in Townsend's Modern Geometry, Vol. I, p. 268. It is not elementary enough to introduce here.

THEOREM (20)—If through any point O, within a triangle ABC, lines AX, BY, CZ are drawn from A, B, C to meet the respectively opposite sides in X, Y, Z; then

Def. A rectilineal figure is said to be of given species, when its angles, and the ratios of the sides forming them, are given.

THEOREM (21)—If a triangle of given species has one corner fixed, and another corner always on a fixed line; then the third corner will always be on a fixed line.

Join CQ, and produce it both ways.

Then, by the construction, As APQ, ABC are simr.

THEOREM (22)—If a triangle, of given species, has one corner fixed, and another corner always on a fixed circle; then the third corner will always be on a fixed circle.

Def. If the opposite pairs of sides of a quadrilateral (defined as on p. 51) are produced to meet, and their points of intersection joined; then the join is called the third diagonal of the quadrilateral; and the figure thus formed is called a complete quadrilateral.

Note The term complete quadrilateral is sometimes used in the following more extended sense-Let there be four indefinite lines, of which no three pass through the same point: they will enclose a four-sided figure (the ordinary quadrilateral) and will have six points of intersection, four of the adjacent sides of the figure, and two others of opposite sides: these six points form three pairs of opposite corners; and there will be three joins of these opposite corners, intersecting in three points. Then the entire figure, consisting of seven lines, intersecting in nine points, is called a complete quadrilateral; and the three joins of the pairs of opposite corners are called its diagonals.

THEOREM (23)—The mid points of the three diagonals of a complete quadrilateral are in one line.

...

P, Q, X are in one line.

mid pts. of YP, YQ, YX are in a || to PQX.

But mid pt. of YP is mid pt. of AC, they are diags. of

CYAP.

And mid pt. of YQ is mid pt. of BD,.. they are diags. of ☐ BYDQ. i.e. the mid pts. of the three diags. AC, BD, XY are in one line.*

* Taken from Taylor's Ancient and Modern Geometry of Conics (p. 255), by permission of the Author.

THEOREM (24)-(Newton-Principia, Book i. Lemma 23) If from two fixed points A, B, lines AX, BY are drawn in fixed directions, and so that AX to BY is a fixed ratio; and if P is taken in XY so that PX to PY is a fixed ratio; then the Locus of P is a fixed line.

.. species of ▲ XOK is fixed.

Draw PL || to OY, to meet XK in L; and join OL.

Then, since OK: KX is fixed;

and KX KL is also fixed;

.. OK: KL is fixed.

..species of ▲ OKL is fixed.

EXERCISES ON BOOK vi.

NOTE-These Exercises are all Theorems to be proved; and depend mainly on the principles of Book vi.

1. If from a point outside a circle, a pair of tangents and a secant are drawn, the quadrilateral formed by joining the points of section to the points of contact, has the rectangles under its opposite sides equal.

2. If two circles touch, a common tangent is a mean proportional between their diameters.

3. CAB, DAB are two triangles on same side of AB; if P is any point in AB; and PX, PY parallels to AC, AD, meet BC, BD respectively ín X, Y; then XY is parallel to CD.

4. The diagonals of a regular pentagon cut each other in extreme and mean ratio.

5. If a radius of a circle is cut in extreme and mean ratio, the greater segment is equal to a side of a regular inscribed decagon.

6. The following group of Theorems are all deducible from Ptolemy's Theorem-vi. Addenda (9) Note (1)—

(1) The distance of any point on the circum-circle of an equilateral triangle, from the farthest corner, is equal to the sum of its distances from the other two

corners.

(2) If the diagonals of a cyclic quadrilateral cut at right angles, then the rectangles under the opposite sides are together double the area of the quadrilateral.

(3) A, B are fixed points on the circumference of a circle; if P is a variable point on the same circumference, and C the mid point of the arc AB, then the ratio PA + PB to PC is constant.

(4) ABCDE is a regular pentagon; if P is any point on the arc AE of its circum-circle, then PA + PC + PE PB + PD.

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NOTE-Apply the Theorem to quads. PABC, PBCD, PBCE.

(5) A similar Theorem to (4) holds for a regular heptagon.

(6) A variable circle goes through the vertex A of a fixed angle, and cuts its sides in X, Y; if the circle also goes through a second fixed point B, then1. AX + m. AY = n. AB;

where 1, m, n are constants whose ratio is determinable.

(7) ABCD is a parallelogram; if a variable circle through A cuts AB, AC, AD in X, Y, Z respectively, then—