PROPOSITION XI. THEOREM. (Eucl. vi. 12.) To find a FOURTH proportional to three given straight Let A, B, C be the three given st. lines. It is required to find a fourth proportional to A, B, C. Take DE, DF, two st. lines making an EDF, and in these make DG=A, GE=B, and DH=C, and through E draw EF || to GH. I. 3. I. 31. VI.2. V. 6. Thus HF is a fourth proportional to A, B, C. Q. E. F. Ex. ABC is a triangle inscribed in a circle, and BD is drawn to meet the tangent to the circle at A in D, at an angle ABD equal to the angle ABC. Show that AC is a fourth proportional to the lines BD, DA, AB. PROPOSITION XII. THEOREM. (Eucl. VI. 8.) In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle and to one another. Let ABC be a right-angled ▲, having ▲ BAC a rt. 4, and from A let AD be drawn to BC. Then must As DBA, DAC be similar to ▲ ABC, and to each other. Forrt. BDA=rt. ▲ BAC, and ▲ ABD= 2 CBA, .. L DAB= 4 ACB. I. 32. ..ADBA is equiangular, and.. similar to ▲ ABC. VI. 4. In the same way it may be shown that ▲ DAC is equiangular, and .. similar to ▲ ABC. Hence ▲ DBA is similar to ▲ DAC. Δ Q. E. D. COR. I. DA is a mean proportional between BD and DC, For BD is to DA as DA is to DC. VI. 4. COR. II. BA is a mean proportional between BC and BD, For BC is to BA as BA is to BD. VI. 4. COR. III. CA is a mean proportional between BC and CD, For BC is to CA as CA is to CD. VI. 4. Q. E. D. Ex. B is a fixed point in the circumference of a circle, whose centre is C; PA is a tangent at any point P, meeting CB produced in A, and PD is drawn perpendicularly to CB. Prove that the line bisecting the angle APD always passes through B. PROPOSITION XIII. PROBLEM. To find a MEAN proportional between two given straight lines. B C Let AB and BC be the two given st. lines. It is required to find a mean proportional between AB and BC. Place AB and BC so as to make one st. line AC, and on AC describe the semicircle ADC. From B draw BD1 to AC, and join AD, CD. I. 11. III. 31. .. DB is a mean proportional between AB and BC. VI. 12, COR. 1. Q. E. F. Ex. 1. Produce a given straight line, so that the given line may be a mean proportional between the whole line and the part produced. Ex 2 Shew that either of the sides of an isosceles triangle is a mean proportional between the base and the half of the segment of the base, produced if necessary, which is cut off by a straight line, drawn from the vertex, at right angles to the equal side. Ex. 3. Shew that the diameter of a circle is a mean proportional between the sides of an equilateral triangle and a hexagon, described about the circle. Ex. 4. From a point A, outside a circle, a line is drawn, cutting the circle in B and C. Find a mean proportional between AB and AC. DEF. IV. Two figures are said to have their sides about two of their angles reciprocally proportional, when, of the four terms of the proportion, the first antecedent and the second consequent are sides of one figure, and the second antecedent and first consequent are sides of the other figure. Thus, in the diagram on the opposite page, the figures AB and BC have their sides about the angles at B reciprocally proportional, the order of the proportion being DB is to BE as GB is to BF. PROPOSITION XIV. THEOREM. Equal parallelograms, which have one angle of the one equal to one angle of the other, have their sides about the equal angles reciprocally proportional. Let AB, BC be equals, having ▲ FBD= 2 EBG. Then must DB be to BE as GB is to BF. Place the Os so that DB and BE are in the same st. line; then must GB and BF also be in one st. line. I. 14. Parallelograms, which have one angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another. Let the sides about the equals be reciprocally proportional, that is, let DB be to BE as GB is to BF. Then must AB=BC. For, the same construction being made, DB is to BE as GB is to BF, |