Mathematics Dictionary

Mathematics Dictionary
Dr. K. G. Shih

Senior High Mathematics

Question Q001

Q001 Draw a circle passing three given points

Outlines

(1) Construction
- Draw a triangle using the points A, B, C.
- Bisect the sides of the triangle.
- Bisectors are concurrent to a point E which is the ex-center.
- Use EA as radius to draw a circle.
- The circle will pass points A, B, C.
(2) Circum-circle of triangle
- a. What is circum-center ?
- b. How to prove that the biscetors of sides are concurrent.

Study subjects : Find equation of the circle passing three points
Method
- a. Substitute 3 points into x^2 + y^2 +a*x + b*y + c = 0.
- b. Solve 3 linear equations of a,b,c.
- c. Use the completing square to find center and radius.

Substitute (0,0), (0,10) and (6,6) into equation we have
- c = 0 ................................................ (1)
- 0 + 100 + 0*a + 10*b + c = 0 or 10*b + c = -100 ...... (2)
- 36 + 36 + 6*a + 6*b + c = 0 or 6*a + 6*b + c = -72 ... (3)
Eliminate c
- substitute (1) into (2) : 10*b = -100 ................ (4)
- substitute (1) into (3) : 6*a + 6*b = -72 ............ (5)
From (4) and (5) find a,b
- From (4), we have b = -10
- From (5), we have 6*a + 6*(-10) = -72 and hence a = -12/6 = -2
Equation of circle : x^2 + y^2 - 2*x - 10*x = 0
Using completing the square
- (x^2 - 2*x + 1 - 1) + (y^2 - 10*x + 25 - 25) = 0
- (x - 1)^2 + (y - 5)^2 = 26
Hence the center is at (1, 5) and radius is r = Sqr(26) = 5.099