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Form 4 Mathematics
Cyclic Quadrilateral
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Investigation 1 Devise a procedure to construct a circle that passes through A and B. Describe your procedure. Justify your construction using known geometric theorems. How many circles can be drawn passing through A and B? Check your answer by clicking the buttons. |
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Investigation 2(a) ABC is a triangle, with no two sides the same length. If P is a point on the same plane such that PAB, PBC and PCA are isosceles triangles, how many position can P lie? Drag P to investigate Check your answer by clicking the button. |
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Investigation 2(b) ABC is an equilateral triangle. If P is a point on the same plane such that PAB, PBC and PCA are isosceles triangles, how many position can P lie? Check your answer by clicking the button. |
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Investigation 2(c) Devise a procedure to construct a circle that passes through three non-collinear points A, B and C. How many circles can be drawn passing through A, B and C? Check your answer by clicking the buttons. |
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ABC is a triangle. The perpendicular bisectors of AB and BC are drawn to meet at D. What is the locus of D when B moves? You can drag the point B yourself to observe the locus or click the "Animate" button. Click the red "X" at the lower-right corner to erase the locus. |
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Circumcircle and circumcentre The circle that passes through the vertices of a triangle ABC is called the circumcircle of the triangle ABC. The centre D of the circle is called the circumcentre. |
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Investigation 4 Using your experience in the above investigations, investigate the possibility of drawing a circumcircle of a given quadrilateral. You may begin by considering special quadrilaterals like rectangle, parallelogram and symmetric trapezium. Record your findings. Does any quadrilateral have a circumcircle? Investigate. If yes, how many. Hypothesize condition(s) under which a circle can be drawn through four non-collinear points. Write down your conjectures with a diagram. If possible, prove your conjectures using known geometric theorems. In your investigation, you may use paper and pencil, compasses or dynamic geometry software. You may also use the following diagram by varying the centre and radius of the circle.
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