given three points, you can always find a circle passing through them all
1. draw lines from a to b to c
2. draw perpendicular bisectors (dashed)
3. the circle center is their intersection point
4. the circle radius is the distance from the center to any of the three points
1. draw lines from a to b to c
2. draw perpendicular bisectors (dashed)
3. the circle center is their intersection point
4. the circle radius is the distance from the center to any of the three points
this works because
• the circle center is by definition, the same distance to all three points
• given a pair of points, the region where the distance to both is the same, is a perpendicular line between them
• the intersection point is therefore where all three are the same
• the circle center is by definition, the same distance to all three points
• given a pair of points, the region where the distance to both is the same, is a perpendicular line between them
• the intersection point is therefore where all three are the same
note that if all three points lie along a line, the circle is undefined! the bisectors would never intersect, so there is no circle passing through all points
(also if you comment about this because you don't read threads I will link you this tweet and make you feel *shame*)
(also if you comment about this because you don't read threads I will link you this tweet and make you feel *shame*)
if we interpret the three points as vertices of a triangle, this circle is usually called the circumcircle
the category of polygons where all points lie in a circle are called cyclic polygons! and so, by definition, all triangles are cyclic polygons
the category of polygons where all points lie in a circle are called cyclic polygons! and so, by definition, all triangles are cyclic polygons
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