HKDSE MathematicsHKDSE Mathematics

Ronald HuiRonald Hui

Tak Sun Secondary SchoolTak Sun Secondary School

18 September 201518 September 2015Ronald HUIRonald HUI

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18 September 201518 September 2015Ronald HUIRonald HUI

18 September 201518 September 2015Ronald HUIRonald HUI

18 September 201518 September 2015Ronald HUIRonald HUI

18 September 201518 September 2015Ronald HUIRonald HUI

18 September 201518 September 2015Ronald HUIRonald HUI

Book 5A Chapter 1Book 5A Chapter 1

Concyclic Points

A, B, C, D and E lie on the same circle.

D

A

B

C

E

D

A

B

C

E

We say A, B, C, D and E are concyclic points.

P Q

Can we always draw a circle passing through two distinct

points P and Q?

Yes! Let us look at some

examples.

C

A

B

How about three non-collinear points A, B and C?

Yes. There is one circle passing

through A, B and C.

Theorem 1.20

and there is one and only one circle that can be drawn passing through them.

Any three non-collinear points are concyclic,

C

A

B

In fact, we have the following theorem:

B

C

A

Step 1

Step 2

Step 3

Draw AB, AC and BC.

Mark their intersection as O. Draw the circle with centre O and radius OA.

ODraw any two of the three bisectors.

We can always draw a circle passing through three non-collinear

points by the following way.

B

C

A

The circle obtained is called the circumcircle of △ABC.

The centre O is called the circumcentre.

O

The radius OA is called the circumradius.

circumradius

circumcentrecircumcircle

The circle passing through A, B and C is obtained.

C

A

B

D

These four points are concyclic.

C

A

B

D

These four points are not concyclic.

Are any four non-collinear points always concyclic?

No

C

A

B

D

ABCD is not a cyclic quadrilateral.

Are any four non-collinear points always concyclic?

No

ABCD is a cyclic quadrilateral.

C

A

B

D

How can we know whether four points are concyclic?

We can test whether four points are concyclic by

one of the following theorems.

Theorem 1.21 (Converse of Theorem 1.9)then A, B, Q and P are concyclic.

P

A B

Q

Abbreviation:converse of s in the same segment

Abbreviation:converse of s in the same segment

If p = q,

p

q

or B + D = 180,

Theorem 1.22 (Converse of Theorem 1.18)

then A, B, C and D are concyclic.If A + C = 180

A

B C

D

Abbreviation:opp. s supp.Abbreviation:opp. s supp.

Theorem 1.23 (Converse of Theorem 1.19)

then A, B, C and D are concyclic.

Abbreviation:ext. = int. opp. Abbreviation:ext. = int. opp.

If q = p,

p

q

A

BC

D

Example:In the figure, AC and BD intersect at K. Prove that A, B, C and D are concyclic.

A

B C

D

70

30100

K

In △ABK,

BAC = 100 30

= 70

∵ BAC = BDC = 70

∴ A, B, C and D are concyclic.

ext. of △

converse of s in the same segment

Refer to the figure. If ABCF is a cyclic quadrilateral, determine whether CDEF is a cyclic quadrilateral.

Follow-up question

B C D

F

E

A

50

11030 FCBDEF ∵

∴ CDEF is not a cyclic quadrilateral.

80

AFE and BCD are straight lines.

8050

= 50∠FCB = 30 + 50

s in the same segment

18 September 201518 September 2015Ronald HUIRonald HUI

江澤民數學題江澤民數學題 (2000)(2000)

江澤民主席在澳門回歸一周年慶典之江澤民主席在澳門回歸一周年慶典之後，到濠江中學參觀。江主席語重心後，到濠江中學參觀。江主席語重心長地對在場的教師說：「我也曾在中長地對在場的教師說：「我也曾在中學教過書，與你們是同行，教師的職學教過書，與你們是同行，教師的職業是非常高尚的。」他興致勃勃地給業是非常高尚的。」他興致勃勃地給大家出了一道幾何題，請大家解答。大家出了一道幾何題，請大家解答。他說，學習幾何能鍛煉一個人的思維他說，學習幾何能鍛煉一個人的思維，解答數學題，最重要的是培養一個，解答數學題，最重要的是培養一個人的鑽研精神，教師對江主席的話報人的鑽研精神，教師對江主席的話報以熱烈的掌聲。以熱烈的掌聲。

18 September 201518 September 2015Ronald HUIRonald HUI

18 September 201518 September 2015Ronald HUIRonald HUI

18 September 201518 September 2015Ronald HUIRonald HUI

18 September 201518 September 2015Ronald HUIRonald HUI

Chapter 1Chapter 1

SQ1: 2/10 (Fri)SQ1: 2/10 (Fri) Revision Ex: 30/9 (Wed)Revision Ex: 30/9 (Wed) Time to work harder please!!!Time to work harder please!!!