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Centres of Triangles in the Coordinate Plane: Paper 1 Challenge

Challenge tier. These questions state a condition, not a method: a vertex hidden behind its orthocentre or in-centre, a triangle given only by its sides, a coordinate condition with more than one answer. Work the standard Paper 1 set first. Answers sit under each question. Detailed solutions can be requested by email.

5 questions · about 50 minutes · answers below each question

All questions are original, written for Math and AI Academy and modelled on past-paper style. They are not reproductions of HKEAA examination questions.

Question 1 · Challenge — runs HKDSE 2021 Paper 2 Q41 backwards: the orthocentre is given and a vertex is recovered, then the in-centre enters with a surd

The coordinates of the points

A

and

B

are

(- 3, 0)

and

(5, 0)

respectively.

C

is a point such that the orthocentre of

△ A B C

H (1, 2)

(a) Find the coordinates of $C$ . (3 marks)

(b) Someone claims that the circumcentre of $△ A B C$ lies on the straight line passing through $C$ and $H$ . Is the claim correct? Explain your answer. (2 marks)

(c) Find the distance between the in-centre and the circumcentre of $△ A B C$ . Give your answer in surd form. (2 marks)

Show answer

(a) $C = (1, 8)$
(b) Correct. $C A = C B$ , so the perpendicular bisector of $A B$ , the line $x = 1$ , passes through $C$ and $H$ ; the circumcentre $(1, 3)$ lies on it.
(c) $5 - 25$

Question 2 · Challenge — new idea relative to the standard set: a general right-angled triangle in letters, and a claim about where its centres can meet

Let

O

be the origin. The coordinates of the points

P

and

Q

are

(p, 0)

and

(0, q)

respectively, where

p

and

q

are positive numbers. Denote the centroid, the circumcentre and the in-centre of

△ O P Q

G

K

and

I

respectively.

(a) Prove that $O$ , $G$ and $K$ are collinear. (2 marks)

(b) Someone claims that $I$ can lie on the straight line $O K$ only when $p = q$ . Is the claim correct? Explain your answer. (2 marks)

(c) Suppose that $p = 8$ and $q = 6$ . Find the area of the triangle with vertices $G$ , $I$ and $K$ . (3 marks)

Show answer

(a) $G = (\frac{p}{3}, \frac{q}{3})$ and $K = (\frac{p}{2}, \frac{q}{2})$ both lie on $y = \frac{q}{p} x$ , which passes through $O$ .
(b) Correct. $I$ always lies on $y = x$ , and $O K$ is $y = \frac{q}{p} x$ ; the two lines coincide only when $p = q$ .
(c) $\frac{1}{3}$

Question 3 · Challenge — modelled on HKDSE 2020 Paper 2 Q40 (a triangle given by its three sides), extended to all four centres and a claim about their order

A triangle is bounded by the straight lines

y = 0

4 x - 3 y = 0

and

4 x + 3 y - 48 = 0

(a) Find the coordinates of the three vertices of the triangle. (2 marks)

(b) Find the coordinates of the orthocentre and of the circumcentre of the triangle. (3 marks)

(c) Someone claims that the in-centre of the triangle lies between its centroid and its orthocentre. Is the claim correct? Explain your answer. (3 marks)

Show answer

(a) $(0, 0)$ , $(12, 0)$ and $(6, 8)$
(b) Orthocentre $(6, \frac{9}{2})$ ; circumcentre $(6, \frac{7}{4})$
(c) Correct. All four centres lie on $x = 6$ ; centroid $(6, \frac{8}{3})$ , in-centre $(6, 3)$ , orthocentre $(6, \frac{9}{2})$ , and $\frac{8}{3} < 3 < \frac{9}{2}$ .

Question 4 · Challenge — a genuine trap: the condition has two answers, and the second part needs every obtuse case, not just one

The coordinates of the points

P

Q

and

R

are

(- 2, 0)

(8, 0)

and

(r, 4)

respectively, where

r

is a constant.

(a) Find all values of $r$ such that the circumcentre of $△ P QR$ lies on $P Q$ . For each value, write down the orthocentre of $△ P QR$ . (3 marks)

(b) Find the range of values of $r$ such that the orthocentre of $△ P QR$ lies outside $△ P QR$ . (4 marks)

Show answer

(a) $r = 0$ or $r = 6$ ; the orthocentre is $R$ itself: $(0, 4)$ or $(6, 4)$ respectively.
(b) $r < - 2$ or $0 < r < 6$ or $r > 8$

Question 5 · Challenge — runs HKDSE 2025 Paper 2 Q41 backwards: the in-centre is given and a vertex is recovered, then the angle bisector property is tested

Let

O

be the origin. The coordinates of the point

Q

are

(0, 12)

P

is a point on the positive

x

-axis such that the in-centre of

△ O P Q

I (3, 3)

(a) Find the coordinates of $P$ . (3 marks)

(b) The straight line $P I$ cuts $O Q$ at $T$ . Find the coordinates of $T$ . (2 marks)

(c) Someone claims that $O T : T Q = O P : P Q$ . Is the claim correct? Explain your answer. (2 marks)

Show answer

(a) $P = (9, 0)$
(b) $T = (0, \frac{9}{2})$
(c) Correct. $O T : T Q = \frac{9}{2} : \frac{15}{2} = 3 : 5$ and $O P : P Q = 9 : 15 = 3 : 5$ .

Continue practising this topic: Try the Paper 1 long questions to build the basics, or move to Paper 2 multiple-choice — also available as Paper 2 Challenge MC.

Where this sits in the syllabus

Recent HKDSE Paper 1 questions give fewer named steps and more single unguided asks, often ending with a claim to check. The skill tested here is translation: turning “the circumcentre lies on a side” into “the angle opposite is 90°” before any calculation starts.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

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