真三弟雅六弟日二十月二年西巴夏
WAH KIU YAT PO
1969
題預習
數學科 (# =)
歐陽海交
MATHEMATICS (23).
Since PE, PQ are tangents from P to A
Similarly,
AP bisects EPQ
Bo bisecta 4 POH
EPQPOH alt.. CHE
AP
BQ
alt.es, paq
Similarly, AQ, BP are the bisectors of alt. GOP, OPF.
AQA BP
AP #BQ
and AQ BP APBQ is a # gram
AQ BQ are the bisectors of two adj. 28 on DNI. Jine GH.
And,
AQB of 2 rt. 48-1 APBQ is a rectangle
ABE PS
diag
of a
Example 7:
Alternate Segment
ABCD is a minor are of a circle such that ABEC. AB and DC.. meet when produced, at P, and, DB is produced to meet the tangent AT at T. Prove that
TP TA
Proof AB=BC
stand on equal chords are equal.
AT is a tangent at a
9 in alt segment
T
This line BC may not
pass thru xit the
intersecting pt. of 20)
TATE
Dare concyclic
đị
(d,)
Zs in same, segment.
sides opp. equal t
LESSON 23 :: CIRCLES
Example 1: (on chords of a circle)
X, Y are the mid-points of the
chords AB, CD of a circle, centre
D; XN, Y are the perpendiculars
from X, Y cb CD, 4B resp. If XN
cuta T at P. prove that OP and XY bisect each other.
Proof: Join OX, OY
** I is the mid-point of the chord AB
DILAB
Since IM LAB (given)
OX // THE?
Similarly, both OY, IN are perp. to CD, "OF // XN
OX // PY," and · · OY/XF
By definition, PYOY 19 therefore a #gram
OP and XY bisect each other
Example 21 (Angle Properties of a circle) The bisectors of the angles ABC, ACB
of ABC intersect at. I and cut
AC, AB at Y,Z resp,; the
circles BIZ, CII mest
again at 1.
Prove that. !
2 YXZ +
BIC=2rt, ≤ 8.
Proofs Join II, with the
notations as shown in
By s in same segment
IXY -,
IXZ = b
cmc2, b, b. (given)
¿YX2—2İXY + « IX2
From 4 IBC, 2BIC + b_+c=2rt. 48.( 2 sum of a
BIC + YXZ = 2
„xample 3; (Coneyclic points)
ABCD 1s a gram
O is a pt. inside ABCD 9.1. ZAOB +4 COD=2 rt Prove that: OBC LODC
coot: Draw AP & DO, BPW CO.
Since the three sides of A ABP are parallel to the three sides.
of a CDO A
But
ABP ADCO
ABCD opp. sides, gram
ABP 4-DOO ASAP
< APB = DOC corr. 45 of SA ZAPB+ZAOB = 2000 + 2A05 #2 rt s.
P, B, O are concyclic Join OP OGAZZI PB
OP # BC &
ad.
a
Example 41 (enords & Ares)
A ABC inscribed in a circle
15 shown. Dis the mid-pt.
Of minor are BC. 0 is a soint on DA sit.
DO=DO
Prove that O is the in-
constr. & proved" proved
s in same segment.
Example 6: ( Contact of circles )
În ▲ Aä, AB ➡p, AC➡q, ~BAC=90° and p>q, 0 is the mid-pt. of BC. Circles are drawn with AB and AC sa diameters. Prove that two circles can be drawn with O as centre to touch each of these circles, and find their radil in terms of a and a
HI • 30= HQ **
Proof: Let E,F be the mid-pt. of AB, AC resp.
Join OE and produce it to cut E at H-K- OF MN.
Where OFAC (mid-point theorem)
AB OH =
(ABAC) Similarly, CM=OF+FM Where OF AB and FFA | AC
* ON = } (AB♣ AG) m From (a) and (b), we have. O
centre of AABC.
roof: Join OC
minar am 83 — minær are DC
Giren
In the same circle, agual ares zublend equal as (at o*)
NOTE:
1.5. UA is the bisector of BAU
(a)
It is required to prove that OC is the bisector of C. In AOAC In a CGD
Exte of a base
isos, a (DO=DC)
But cama Bi Bi ( 5. in same set.
1.6. 00 is the bisector of BCA- from (a) and (b), we conclude that
0 is the in-centre of A ABC
Pis
xample 5: ABC is an equilateral
inscribed in a circle. any point on minor are BC. Prove that PAPB+ RE
Proof, Produce BP to D
such that PD PC
In ACPD: PCPD constr.
CPD BAC - 60° (Ext. of cyclic quad. PCD is equilateral
BCD:
Consider 8 APC
AC = BC ACPBCD (=60°+*
ACAPRABOC AP
sides of equi a PCD
Since BDBP+PD-BP+ PC
AP BP+FC
Fence, with centre 0, radius Of =2(pra) we can draw one such circle.
Secondly, consider now, ON and OK
OK = EX - EO -(AB~ AC)=(p-q)
ON OF NF = ( AB ~ AG) = 3( p−q) Hence, with centre 0, radius OK= £(p=0 we can draw another req'd cirei
“As shown in figure (to the left)
(a) AABC is said to be
circumscribed about @ o (b) Quad LNP is said to be
inscribed in 0.0
(c) 0 0 is said to be
circumscribed about IMNP or inscribed in 4 ABC.
Flints & Art To Ex. 24
PACOF
& FCD⇒ 4 ACB
SIS AM
Ikram XQAD=How QPBY
As shown in figure, in a ABC, ROH BC. PO HAB ABP=▲ ABQ = LARETARIQ
DACR = & ARE + AROC
ARRC=A RRB
Hence
SAB⋅P = SACR
AD is a median, the orem.
Apollonius
AC*= 2(AD*+ BD*) AB= 2AD- AC" 4
AD
AB'
#BD"
= A D + CD*+ » BD
We may take a point. E on AP such that PE #PB, Hence, prove that 4 BPC = ▲ ABE ( ABPE is equilateral), then POW AB-
Example 6: (tangents)
As shown in fig., A, B are centres of two circles Low- ing the parallel lines GOH and EPF, the common tangent
PQ touches each circle as shown. Prove that
(a) APBQ is a rectangle,
(b) PQ AB
APU QC
AQCP is a ligom area of #gram AQCP is
AQ• AB
AD AP
PC AB
AD QC
Hence
QA AB
AD AP
二星日八月四年九六九一插公年八十五國民
育華僑華 育教
1969 TUP
英文科
(廿三)
桂
a) (1) and
ENGLISH (23)
Answer to Paper XXI
b) (iv) incentive.
o) (11) outside
d) (111) fairness
(1) choices.
f) (11) worldwide.
a)(11) There are several small ways in which
examinations could be improved.
5) (111) Industry and commerce depend upon. examination certificates.
c) (iv) Since the war many countries have tried to
give more candidates more freedom in deciding what they wish to be examined in
Paper XXIII/
Hend the following carefully and make a pre using your own words as far as possible.
The charm of the big store spring a from the fact that it is more than a universal shop it is a. universal exhibition, always open and always free. In material matters it keeps you up to date with the progress of the world, and it is therefore an education for everybody who enters. It is also a tonic and a stirrer of the imagination and of ambition in the too sluggish breast. You may, and generally do, go into a store to buy, but that is only part of your aim. You go in order to watch human nature, to see what other people are buying, to compare your tastes with other people's tastes, and to criticize both yours and theirs. You go, further, to see what you would buy if you could afford to buy it. And if you emerge from the store disgusted with your own clothes, or your own furniture, or your own gadgets and dodges for getting the most out of your daily home at the least possible cost so much the ambition is then born.
the better, for laudable
There are individuals who say that they hate. shopping. Of them it is to be said, either that they do not know what shopping is, or they have not acquired the technique of shopping, or they are blind and deaf to the great spectacle of the world, Fortunately, the number of haters of shopping has creditably diminished within the last few decades. The big stores by their insidious arts have seen to thật. The big stores have transformed shopping into pastime – perhaps dangerous, but a pastime.
me may wonder sometimes how if you buy a few shillings' worth of firewood from them they can afford, besides delivering the sticks at your door in a five- ton_motor-lorry, to offer you gratis a clubroom, an information bureau, e writing-room, note-paper, and a sort of permanent exhibition. But the big stores have thought it carefully out. The thing does indeed pay, and their dividends prove it.
We need not regret the departed picturesqueness
of the marts of old- Tyre, Sidon, Rome, Venice. No⠀⠀ mart of old could ever have rivalled in picturesqueness, in colour, in richea, in variety, and in fascination, the big stores of the great cities of today. No mart of old was ever fed by so many ships from such distant ports as the big stores of our era,
It seems certain that when this civilisation has fallen into ruins, the big stores, which we now take for granted will be presented to the historical students of a few thousand years hence as incredible marvels romance, vitality and enterprise, and, that those students will sigh because for them the age of miracion is long past.
EXERCISE 23
1) A triangle ABC is inscribed in a circle, and the bis- ectors of the ce meet the-0 at X,Y,Z. Show that: the as of a XIZ are resp. 90°-90 90-
2) Two circles meet at A and B, AC,AD are diameters
each circle. Prove that C,B,D are collinear.
3) Two circles meet at A,B. CD is a common tangent to two
circles. Prove that CAD + CBD=180*
4) 4 ABC is inscribed in a circle. O is the orthocentre
of 4 ABC. The three altitudes AD, BE, CF are produced to meet the circle again at G,H,K resp. Prove that Q is also the incentre of a GHK,
5) Two equal chords AB,CD meet, when produced, at G.
Frove that BGDG.
6) ABCD is a cyclic quad. If H‚N,P,Q are the centres of
four circles inscribed in As ABC.
1
that HNFQ is a rectangle. ARDE
ACD, BCD Prove
7) The diagonals of cyclic quad ABCD cut each other at
rts at P. Prove that the from P to BC blaacts, when produced,. AD. (Brachmegupta theorem).
8) AOB, COD are two 1 diameters of a circle. Two chorda:
CP, CQ cut AB at H, K, prove that H,K,Q,P, are concyclic
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