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NOTE. Propositions B, C, and D do not occur in Euclid, but were added by Robert Simson, who edited Euclid's text in 1756.
Prop. D is usually known as Ptolemy's theorem, and it is the particular case of the following more general theorem :
The rectangle contained by the diagonals of a quadrilateral is less than the sum of the rectangles contained by its opposite sides, unless a circle can be circumscribed about the quadrilateral, in which case it is equal to that sum.
1. ABC is an isosceles triangle, and on the base, or base produced, any point X is taken : shew that the circumscribed circles of the triangles ABX, ACX are equal.
2. From the extremities B, C of the base of an isosceles triangle ABC, straight lines are drawn perpendicular to AB, AC respectively, and intersecting at D: shew that the rectangle BC, AD is double of the rectangle AB, DB.
3. If the diagonals of a quadrilateral inscribed in a circle are at right angles, the sum of the rectangles contained by the opposite sides is double the area of the figure.
4. ABCD is a quadrilateral inscribed in a circle, and the diagonal BD bisects AC: shew that the rectangle AD, AB is equal to the rectangle DC, CB.
5. If the vertex A of a triangle ABC is joined to any point in the base, it will divide the triangle into two triangles such that their circumscribed circles have radii in the ratio of AB to AC.
6. Construct a triangle, having given the base, the vertical angle, and the rectangle contained by the sides.
7. Two triangles of equal area are inscribed in the same circle: shew that the rectangle contained by any two sides of the one is to the rectangle contained by any two sides of the other as the base of the second is to the base of the first.
8. A circle is described round an equilateral triangle, and from any point in the circumference straight lines are drawn to the angular points of the triangle : shew that one of these straight lines is equal to the sum of the other two.
9. ABCD is a quadrilateral inscribed in a circle, and BD bisects the angle ABC: if the points A and C are fixed on the circumference of the circle and B is variable in position, shew that the sum of AB and BC has a constant ratio to BD.
THEOREMS AND EXAMPLES ON BOOK VI.
ON HARMONIC SECTION.
1. To divide a given straight line internally and externally so that its segments may be in a given ratio.
Let AB be the given st. line, and L, M two other st. lines which determine the given ratio. It is required to divide AB internally and externally in the ratio L:M. Through A and B draw any two par? st. lines AH, BK.
From AH cut off Aa equal to L, and from BK cut off Bb and Bb' each equal to M, Bb' being taken in the same direction as Aa, and Bb in the opposite direction.
Join ab, cutting AB in P; join ab', and produce it to cut AB externally at Q. Then AB shall be divided internally at P and externally at Q, so that
AP : PB=L:M. and
AQ: QB=L:M. The proof follows at once from Euclid vi. 4. NOTE. The solution is singular; that is, only one internal and one external point can be found that will divide the given straight line into segments which have the given ratio.
DEFINITION. A finite straight line is said to be cut harmonically when it is divided internally and externally into segments which have the same ratio.
Thus AB is divided harmonically at P and Q, if
AP : PB=AQ : QB.
PA : ĀQ=PB : BQ; from which it is seen that if P and Q_divide AB internally and externally in the same ratio, then A and B divide PQ internally and externally in the same ratio; hence A and B are harmonic conjugates of P and Q.
Example. The base of a triangle is divided harmonically by the internal and external bisectors of the vertical angle : for in each case the segments of the base are in the ratio of the other sides of the triangle. [Euclid vi. 3 and A.]
Obs. We shall use the terms Arithmetic, Geometric, and Harmonic Means in their ordinary Algebraical sense.
1. If AB is divided internally at P and externally at Q in the same ratio, then AB is the harmonic mean between AQ and AP.
For, by hypothesis, AQ : QB=AP : PB; .., alternately, AQ : AP=QB: PB, that is,
AQ : AP =AQ - AB : AB - AP; :: AP, AB, AQ are in Harmonic Progression. 2. If AB is divided harmonically at P and Q, and O is the middle point of AB;
then OP. OQ=OA?.
For since AB is divided harmonically at P and Q,
:: AP : PB=AQ : QB; :: AP - PB : AP + PB=AQ -QB : AQ+QB, or,
20P : 20A=20A : 200 ;
:: OP. OQ=OA?. Conversely, if OP. OQ=OAP, it may be shewn that
AP : PB=AQ : QB;
3. The Arithmetic, Geometric and Harmonic means of two straight lines may be thus represented graphically. In the adjoining figure, two tan
H gents AH, AK are drawn from any external point A to the circle PHQK; HK is the chord of contact, and the st. line joining A to the centre O cuts
В о the Oce at P and Q.
Then (i) AO is the Arithmetic mean between AP and AQ: for clearly
Ex. 1, p. 251. :. AB is cut harmonically at P and Q. Ex. 2, p. 385. That is, AB is the Harmonic mean between AP and AQ. And from the similar triangles OAH, HAB,
OA : AH=AH : AB, :: AO . AB=AH2;
VI. 17. .. the Geometric mean between two straight lines is the mean proportional between their Arithmetic and Harmonic means.
4. Given the base of a triangle and the ratio of the other sides, to find the locus of the vertex.
Let BC be the given base, and let BAC be any triangle standing upon it, such that BA : AC=the given ratio. It is required to find the locus of A.
Bisect the L BAC internally and B externally by AP, AQ. Then BC is divided internally at P, and externally at Q, so that BP : PC=BQ : QC=the given ratio ;
P and Q are fixed points. And since AP, AQ are the internal and external bisectors of the
the L PAQ is a rt, angle ;
EXERCISE. Given three points B, P, C in a straight line : find the locus of points at which BP and PC subtend equal angles.
DEFINITIONS. 1. A series of points in a straight line is called a range. If the range consists of four points, of which one pair are harmonic conjugates with respect to the other pair, it is said to be a harmonic range.
A series of straight lines drawn through a point is called a pencil.
The point of concurrence is called the vertex of the pencil, and each of the straight lines is called a ray.
A pencil of four rays drawn from any point to a harmonic range
is said to be a harmonic pencil. 3. A straight line drawn to cut a system of lines is called a transversal.
4. A system of four straight lines, no three of which are concurrent, is called a complete quadrilateral.
These straight lines will intersect two and two in six points, called the vertices of the quadrilateral; the three straight lines which join the opposite vertices are diagonals.
THEOREMS ON HARMONIC SECTION. 1. If a transversal is drawn parallel to one ray of u harmonic pencil, the other three rays intercept equal parts upon it: and conversely.
2. Any transversal is cut sharmonically by the rays of a harmonic pencil.
3. In a harmonic pencil, if one ray bisect the angle between the other pair of rays, it is perpendicular to its conjugate ray: Conversely, if one pair of rays form a right angle, then they bisect internally and externally the angle between the other pair.
4. If A, P, B, Q and a, p, b, q are harmonic ranges, one on each of two given straight lines, and if Aa, Pp, Bb, the straight lines which join three pairs of corresponding points, meet at S; then will Qq also pass through S.
5. If two straight lines intersect at A, and if A, P, B, Q and A, p, b, q are two harmonic ranges one on each straight line (the points corresponding as indicated by the letters), then Pp, Bb, Qq will be concurrent : also Pą, Bb, Qp will be concurrent.
6. Use Theorem 5 to prove that in a complete quadrilateral in which the three diagonals are drawn, the straight line joining any pair of opposite vertices is cut harmonically by the other two diagonals.