Ex. 1339. Draw a circle (radius about 3 in.); take four points A, B, C, D upon it. By measurement, find the sum of the angles BAD, BCD; also of the angles ABC, ADC.

THEOREM 12.

The opposite angles of any quadrilateral inscribed in a circle are supplementary.

PQRS is a quadrilateral inscribed in PQR*.

(1) < PQR + 4 PSR = 2 rt. ≤ s,

Data

To prove that

Construction

Proof

* The two figures represent the two cases in which the centre is (i) inside,

(ii) outside the quadrilateral. The same proof applies to both.

Ex. 1340. From the given angles, find all the

angles in fig. 257.

Ex. 1342. The side PQ of a quadrilateral PQRS, inscribed in a circle, is produced to T.

fig. 257.

Prove that the exterior LRQT=the interior opposite & PSR.

Ex. 1343. If a parallelogram can be inscribed in a circle, it must be a rectangle.

Ex. 1344. If a trapezium can be inscribed in a circle, it must be isosceles.

8

Ex. 1345. The sides BA, CD of a quadrilateral ABCD, inscribed in a circle, are produced to meet at O; prove that ▲ OAD, OCB are equiangular.

Ex. 1346. ABCD is a quadrilateral inscribed in a circle, having LA=60°; O is the centre of the circle. Prove that

LOBD+LODB=4CBD+4CDB.

Ex. 1347. What is the relation between the angles subtended by a chord at a point in its minor arc, and at a point in its major arc?

Ex. 1348. Draw a quadrilateral ABCD, having A+ ▲ C=180°. Draw a circle to pass through ABC; notice whether it passes through D.

THEOREM 13.

[CONVERSE OF THEOREM 12.]

If a pair of opposite angles of a quadrilateral are supplementary, its vertices are concyclic.

Data The 4s ABC, ADC of the quadrilateral ABCD are supple

mentary.

To prove that

A, B, C, D are concyclic.

Construction Draw to pass through A, B, C.

It must be shown that this passes through D.

Proof If ABC does not pass through D, it must cut AD (or AD produced) in some other point D'.

But this is impossible, for one of the s is an exterior

of ▲ DD'C, and the other is an interior opposite of the same A.

Hence

ABC must pass through D, i.e. A, B, C, D are concyclic.

Q. E. D.

DEF. If a quadrilateral is such that a circle can be circumscribed round it, the quadrilateral is said to be cyclic.

Ex. 1349. BE, CF, two altitudes of AADC, intersect at H. Prove that AEHF is a cyclic quadrilateral.

Ex. 1350. the base BC. Ex. 1351.

ABC, DBC are two congruent triangles on opposite sides of Under what circumstances are A, B, C, D concyclic?

ABCD is a parallelogram. A circle drawn through A, B, cuts AD, BC (produced if necessary) in E, F respectively. Prove that E, F, C, D are concyclic.

Ex. 1352. ABCD is a quadrilateral inscribed in a circle. DA, CB are produced to meet at E; AB, DC to meet at F. Prove that, if a circle can be drawn through AEFC, then EF is the diameter of this circle; and BD is the diameter of ABCD.

Ex. 1353. The straight lines bisecting the angles of any convex quadrilateral form a cyclic quadrilateral.

For further exercises on the subject-matter of the above section see end of section IX.

SECTION VII.

CONSTRUCTION OF TANGENTS.

Ex. 1354. Stick two pins into the paper 2 in. apart at A and B; place the set-square on the paper so that the sides containing the 60° are in contact with the pins; mark the point where the vertex of the angle rests. Now slide the set-square about, keeping the same two sides against the pins, and plot the locus of the 60° vertex. What is the locus? are A, B points in the locus? Complete the circle,

and measure the angle subtended by AB at a

point in the minor arc.

fig. 260.

B

Ex. 1355. Repeat the experiment of Ex. 1354 with the 30° vertex.
Repeat the experiment of Ex. 1354 with the 90° vertex.

Ex. 1356.

Ex. 1357. a right angle?

What is the locus of points at which a given line subtends

Ex. 1358. O is the centre of a circle and Q is a point outside the circle. Construct the locus of points at which OQ subtends a right angle. Find two points A, B on the first circle, so that LOAQ=LOBQ=90°. Prove that QA is a tangent to the first circle.

To draw tangents to a given circle ABC from a given point T outside the circle.

B

fig. 261.

T

Construction Join T to O, the centre of O ABC.

On OT as diameter describe a

cutting the given circle

These lines are tangents.

Join OA, OB.

Since OT is the diameter of OAT,

..LOAT is a right angle,

.. AT, being to radius OA, is the tangent at A.
Similarly BT is the tangent at B.

Ex. 1359. Draw tangents to a circle of radius 2 ins. from a point lin. outside the circle; calculate and measure the length of the tangents.

Ex. 1360. Draw a circle of radius 3 cm. and mark a point T distant 7 cm. from the centre. Find where the tangents from T meet the circle (i) by the method of p. 240, (ii) as above. Calculate the length of the tangents, and ascertain which method gives greater accuracy.

Ex. 1361. Find the angle between the tangents to a circle from a point whose distance from the centre is equal to a diameter.

Ex. 1362. Through a point 2 in. outside a circle of radius 2 in. draw a line to pass at a distance of 1 in. from the centre. Measure and calculate the part inside the circle.