MISCELLANEOUS EXERCISES

1. If two triangles have one angle of the one equal to one angle of the other, and a second angle of the one supplementary to a second angle of the other, then the sides about the third angles are proportional.

2. AE bisects the vertical angle of the triangle ABC and meets the base in E; show that if circles are described about the triangles ABE and ACE, their diameters are to each other in the same ratio as the segments of the base.

3. Two circles touch internally at O; AB a chord of the larger circle touches the smaller at C which is cut by the lines OA, OB at P and Q; show that OP: OQ = AC: CB.

4. If two triangles have their sides parallel in pairs, the straight lines joining their vertices meet in a point, or are parallel.

5. If any two similar polygons have three pairs of corresponding sides parallel, the straight lines joining the corresponding vertices meet in a point or are parallel.

6. If A, B, C, D are any four points on a circle and E, F, G, H are the mid-points of the arcs AB, BC, CD, DA, respectively, prove that the straight lines EG and FH are at right angles.

7. The sum of the perpendiculars drawn from any point within an equilateral triangle on the three sides is invariable.

8. Prove that the straight lines which trisect one angle of a triangle do not trisect the opposite side.

9. That part of any tangent to a circle which is intercepted between tangents at the extremities of a diameter is divided at the point of contact into segments such that the radius of the circle is a mean proportional between them.

10. If two chords AB and AC, drawn from a point A on a circle ABC, are produced to meet the tangent at the other extremity of the diameter through A, in the points D and E respectively, show that the triangle AED is similar to the triangle ABC.

11. On a circle of which AB is a diameter take any point P. Draw PC and PD on opposite sides of AP and equally inclined to it, meeting AB at C and D. Prove AC: BC AD: BD.

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1. DEFINITIONS.

SUMMARY OF CHAPTER III

(1) To Measure to find out by experiment how many times a given magnitude will contain a chosen unit. § 211.

(2) Multiple of a Given Magnitude-a magnitude which will contain that magnitude an integral number of times. § 211.

(3) Measure of a Given Magnitude

a magnitude which is con

tained in that magnitude an integral number of times. § 211. (4) Commensurable Magnitudes — such as can be measured with a common unit. § 213.

(5) Incommensurable Magnitudes - such as cannot be measured by
any common unit. § 213.

(6) Ratio of Two Quantities — their relative magnitude, i.e. how
many times one is as great as the other.
§ 216.

(7) Ratio of Two Commensurable Magnitudes - the ratio of their
numerical measures by a common unit. § 220.

(8) Ratio of Two Incommensurable Magnitudes the limit which the ratio of their approximate measures by a common unit approaches, as this unit is indefinitely diminished. § 229. (9) Limit of a Variable Quantity — a fixed quantity to which the variable approaches nearer than for any assignable difference, though it cannot be made absolutely identical with it. (10) Proportion · a statement of the equality of two ratios. § 232. (11) Continued proportion, mean proportional, third proportional. See § 236.

§ 227.

(12) Mutually Equiangular Polygons - those having their angles equal, each to each, and in the same order. § 247. (13) Similar Polygons-two polygons which have the same number of sides, are mutually equiangular, and have their pairs of corresponding sides proportional. § 248.

(14) Radical Axis of Two Intersecting Circles - the line of their common chord. § 263.

(15) Coaxial System of Circles - the circles through two fixed points.

§ 263.

(16) Division in Extreme and Mean Ratio — division of a line-segment into two parts, such that one of them is a mean proportional between the whole segment and the other part. § 274.

(17) Harmonic Division-division of a line-segment internally and externally in the same ratio. § 283.

2. POSTULATES.

(1) If P and Q are any two equal magnitudes, and R is a third magnitude of the same kind, then the ratio of P to R is equal to the ratio of Q to R, i.e. if P= Q, then P: R= Q: R; and conversely, if P and Q are such that P: R= Q: R, then P= Q. (Postulate 6.) § 218.

(2) If P and Q are two unequal magnitudes, and R is a third magnitude of the same kind, then the ratio of P to R is greater or less than the ratio of Q to R, according as P is greater or less than Q. (Postulate 7.) § 218.

3. PROBLEMS.

(1) To divide a given line-segment into any required number of equal parts. $269.

(2) To divide a given line-segment into parts proportional to other given line-segments. § 270.

(3) To divide a given line-segment internally or externally in a given ratio. § 271.

(4) To find a fourth proportional to three given line-segments. § 272.

(5) To find a mean proportional between two given line-segments. § 273.

(6) To divide a given line-segment in extreme and mean ratio. § 275. (7) Upon a given line-segment to construct a polygon similar to a given polygon, and such that the given line-segment shall be homologous to a given side. § 276.

(8) To find the locus of a point whose distances from two fixed points are in a constant ratio different from unity. § 282.

4. THEOREM ON LIMITS.

If there are two variable quantities dependent on the same quantity in such a way that they remain always equal while each approaches a limit, then their limits are equal. § 230.

5. THEOREMS ON PROPORTION.

(1) If four numbers are in proportion, the product of the extremes equals the product of the means. § 233.

(2) If a b c : d, then by inversion bad: c, and by alternation

a:cb: d. § 234.

(3) If a b = c: d, then by composition a + b:b

=c+d:d, by division ab:bcd: d, and by composition and division a+b: a b = c +d: c - d. § 237.

(4) If three terms of one proportion are equal, respectively, to the three corresponding terms of another, their fourth terms must be equal. § 238.

(5) If any number of ratios are equal, then the sum of all the antecedents is to the sum of all the consequents as any one antecedent is to its consequent. § 240.

6. THEOREMS ON THE SIMILARITY OF TRIANGLES.

(1) Two triangles are similar if the three angles of one are equal, respectively, to the three angles of the other. § 249.

(2) If two triangles have two angles of one equal, respectively, to two angles of the other, the triangles are similar. § 250.

(3) If two triangles have an angle of one equal to an angle of the other, and the sides including these angles proportional, the triangles are similar. § 251.

(4) If the ratios of the three sides of one triangle to the three sides of another, two and two, are equal, the triangles are similar. § 253.

7. THEOREMS ON CHORDS OF CIRCLES.

(1) If two chords of a circle intersect within the circle, the product of the segments of the one equals the product of the segments of the other. § 258.

(2) If through a fixed point within a circle any chord is drawn, the product of its segments is the same whatever its direction. § 259.

(3) Either segment of the least chord that can be drawn through a fixed point within a circle is a mean proportional between the segments of any other chord drawn through that point. § 260. (4) A perpendicular drawn from any point of a circle to a diameter is a mean proportional between the segments into which it divides the diameter. § 256.

(5) If two chords of a circle, when produced, intersect without the circle, the product of the segments of the one equals the product of the segments of the other.

§ 261.

(6) If a tangent and a secant of a circle intersect, the tangent is a mean proportional between the whole secant and its external segment. § 262.

(7) If from any point on the common chord of two intersecting circles produced, tangents are drawn to the circles, the lengths of these tangents are equal. § 263.

8. MISCELLANEOUS THEOREMS.

(1) A straight line parallel to one side of a triangle divides the other
two sides in the same ratio, and conversely. §§ 242, 244.

(2) The bisector of an angle of a triangle divides the opposite side
into segments proportional to the adjacent sides. § 245.
(3) If two similar polygons are divided into triangles by the diagonals
drawn from a pair of homologous vertices, these triangles are
similar each to each and are similarly placed, and conversely.
§ 252.

(4) The perimeters of two similar polygons are in the same ratio as
any two corresponding sides. § 254.

(5) If in a right triangle the perpendicular is drawn from the vertex of the right angle to the hypotenuse: (1) the two triangles thus formed are similar to each other and to the whole triangle; (2) the perpendicular is a mean proportional between the segments of the hypotenuse; (3) each side of the triangle is a mean proportional between the hypotenuse and the segment adjacent to that side. § 255.

(6) In equal circles, angles at the centre are in the same ratio as the arcs subtending them. § 266.

(7) If the sides of a triangle are cut by any straight line, the product of the ratios of their segments taken in order equals unity, and conversely (Menelaus's theorem). §§ 277, 278.

(8) If two triangles are so situated that the straight lines joining the pairs of corresponding vertices are concurrent, then the points of intersection of the three pairs of corresponding sides are collinear, and conversely (Desargues's theorem). §§ 279, 280. (9) If through any point straight lines are drawn to the vertices of a triangle, intersecting the sides, the product of the ratios of the segments of the sides taken in order equals unity. § 281. (10) If any line-segment AB is divided harmonically at C and D, the line-segment CD is also divided harmonically at A and B. § 284.