CirClever

Geometry Level 3

For which of the following sets of points, does there exist a circle that passes through all of them?

(0,5),(0,0),(9,2) (1,2),(0,1),(2,3) (2,6),(8,12),(5,9) (5,1),(9,-11),(12,-20) (-1,-1),(5,5),(7,7) (1,2),(2,-5),(0,9)

Jess McAllister Alicando by Jess McAllister Alicando , 21, Philippines

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2 solutions

If y 2 − y 1 x 2 − x 1 ≠ y 2 − y 3 x 2 − x 3 \dfrac {y_2 - y_1}{x_2 - x_1} \neq \dfrac {y_2 - y_3}{x_2 - x_3} , that is if the tree points are not in a line, a circle will pass through them. (0,5),(0,0),(9,2) are such points, since first to points are on y-axis, last one is not. .

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Deepak Kumar
Jan 30, 2016

Simply determine which set doesn't represent collinear points i.e forms a triangle and hence the circumcircle of the triangle must pass through the three points.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

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