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In the diagram, the points are evenly spaced vertically and horizontally. A segment $AB$ is drawn using two of the points, as shown. Point $C$ is chosen to be one of the remaining $18$ points. For how many of these $18$ possible points is triangle $ABC$ isosceles?

 Dec 22, 2023
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There are two cases to consider:

 

Case 1: C is on the horizontal line through A or the vertical line through B.

 

Without loss of generality, suppose C is on the horizontal line through A. If AB is a horizontal segment, then △ABC cannot be isosceles.

 

Otherwise, AC=AB if and only if C is one of the two points on this horizontal line that are the same distance from A as B is.

 

The same argument applies if C is on the vertical line through B. Thus, there are at most 2+2=4 such points C.

 

Case 2: C is not on the horizontal line through A or the vertical line through B.

 

Then △ABC is isosceles if and only if C lies on the perpendicular bisector of AB. There are 4 such points.

 

Hence, there are 4+4=8​ such points C.

 Dec 24, 2023

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