HKDSE Mathematics Ronald Hui Tak Sun Secondary School

18 September 2015Ronald HUI Missing Homework SHW1-A1, SHW1-B1 SHW1-A1, SHW1-B SHW1-C1 SHW1-C1 1, 8, 9, 10, 12 1, 8, 9, 10, 12 SKY book $ SKY book $ 5, 8, 9, 10, 18, 5J7 5, 8, 9, 10, 18, 5J7 Summer Holiday Homework Summer Holiday Homework 25 Sep (Fri) 25 Sep (Fri)

18 September 2015Ronald HUI

18 September 2015Ronald HUI

18 September 2015Ronald HUI

18 September 2015Ronald HUI

18 September 2015Ronald HUI

Book 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. O Draw 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 circumcentre circumcircle 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 B C 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  30  100  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 CD F E A 50  110  30  ∵ ∴ CDEF is not a cyclic quadrilateral. AFE and BCD are straight lines. 80  50  = 50  ∠ FCB = 30  + 50   s in the same segment

18 September 2015Ronald HUI

江澤民數學題 (2000) 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 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 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。 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 人的鑽研精神，教師對江主席的話報 以熱烈的掌聲。 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 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 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。 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 人的鑽研精神，教師對江主席的話報 以熱烈的掌聲。 18 September 2015Ronald HUI

18 September 2015Ronald HUI

18 September 2015Ronald HUI

18 September 2015Ronald HUI Chapter 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!!!