THEOREM 7 (i).

If AB, CD, two chords of a circle, intersect at a point P inside the circle, then PA. PB = PC. PD.

THEOREM 7 (ii).

If AB, CD, two chords of a circle, intersect at a point P outside the circle, then PA . PB

=

PC. PD.

NOTE. Theorems 7 (i) and 7 (ii) are really two different cases of the same theorem; notice that the proofs are nearly identical.

G. S. II.

22

Ex. 1723.

If PT is a tangent to a circle

and AB a chord of the circle passing through

P, then PT2 = PA. PB. (See fig. 321.)

To calculate the area of the rectangle PA. PB in ïv. 7 (ii).

Use the fact that

PA. PB = PT2 = OP2 - OT2.

Ex. 1724.

What becomes of IV. 7 when P is a point on the circle? When P is the centre?

fig. 321.

Ex. 1725. Calculate (and check graphically) the areas of the rectangles contained by the segments of chords passing through P when (i) r=5 in., OP=3 in., (ii) r=5 cm., OP=13 cm., (iii) r=3.7 in., OP=2·3 in., (iv) r=2.9 in., OP=3.3 in.

Ex. 1726. Find an expression for the areas in Ex. 1725, ↑ being the radius, and d the distance OP (i) when d<r, (ii) when d>r. Explain fully.

Ex. 1727. Draw two straight lines APB, CPD intersecting at P; make PA=4 cm., PB 6 cm., PC=3 cm. Describe a circle through ABC, cutting CP produced in D. Calculate PD, and check by measurement.

What would be the result if the exercise were repeated with the same lengths, but a different angle between APB, CPD?

Ex. 1728. From a point P draw two straight lines PAB, PC; make PA=4 cm., PB=9 cm., PC-6 cm. Describe a circle through ABC; let it cut PC again at D. Calculate PD, and check by measurement.

Ex. 1729. APB, CPD intersect at P; and the lengths PA, PB, PC, PD are so chosen that PA. PB PC. PD. Prove that A, B, C, D are concyclic. (Draw through ABC; let it cut CP produced in D'.) Make up a numerical instance, and draw a figure. What relation does this exercise bear to IV. 7 (i)?

Ex. 1730. State and prove the converse of iv. 7 (ii).

Ex. 1731.

P is a point outside a circle ABC and straight lines PAB, PC are drawn (A, B, C being on the circle); prove that, if PA. PB PC3, PC is

the tangent at C.

[Use reductio ad absurdum.]

Ex. 1732.

ABC is a triangle right-angled at A; AD is drawn perpen

dicular to BC; prove that AD2= BD. DC.

[Produce AD to cut the circumcircle of ▲ ABC.]

Ex. 1733. If the common chord of two intersecting circles be produced to any point T, the tangents to the circles from T are equal to one another.

Ex. 1734. The common chord of two intersecting circles bisects their common tangents.

Ex. 1735. The altitudes BE, CF of a triangle ABC intersect at H, prove that

(i) BH. HE=CH.HF, (ii) AF. AB=AE.AC, (iii) BH.BE=BF . BA.

Ex. 1736. Two circles intersect at A, B; T is any point in AB, or AB produced; TCD, TEF are drawn cutting the one circle in C, D, the other in E, F. Prove that C, D, E, F are concyclic.

Ex. 1737. ABC is a triangle right-angled at A; AD is an altitude of the triangle. Prove that ▲ ABD, CDA are equiangular. Write down the three equal ratios; and, by taking them in pairs, deduce the corresponding rectangle properties.

=

xb, then x is

DEF. If x is such a quantity that a: x called the mean proportional between a and b.

Ex. 1738. Prove that, if x is the mean proportional between a and b, x2=ab.

To find the mean proportional between two given straight lines.

Let a, b be the two given straight lines. Construction Draw a straight line PQ.

Proof

From PQ cut off PR = a, and RS = b.

On PS as diameter describe a semicircle.

Through R draw RT to PS to cut the semicircle at T.
Then RT (x) is the mean proportional between a, b.

.. x is the mean proportional between a and b.

Ex. 1740. Prove the above construction by completing the circle, and

producing TR to meet the circle in T'.

proportionals

Ex. 1741. (On inch paper.) Find graphically the mean between (i) 1 and 4, (ii) 1 and 3, (iii) 1·5 and 2·5, (iv) 1·3 and 1·7. Check by calculation.

α х

NOTE.

If

x2 = ab, and therefore x= х b'

√ab; thus the mean proportional between two numbers is the square root of the product.

Ex. 1742. (On inch paper.) Find the square roots of (i) 2, (ii) 3, (iii) 6, (iv) 7.

[Find the mean proportionals between (i) 1 and 2, (iii) 2 and 3.]

Ex. 1743. Draw a triangle; and construct an equivalent rectangle. [What is the formula for the area of a triangle ?]