Question 1199525
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The response from the other tutor shows a basic standard geometric proof; it does not use coordinate geometry.<br>
Coordinate geometry is a tool with which geometric proofs are sometimes far easier than with Euclidean geometry.<br>
The circumcenter of any triangle lies on the intersection of the perpendicular bisectors of any two sides of the triangle.<br>
Side AB of the given triangle is a vertical segment with midpoint (c,(d+e)/2), so the equation of the perpendicular bisector of AB is the horizontal line y=(d+e)/2.<br>
Side BC of the given triangle is a horizontal segment with midpoint ((c+f)/2,e), so the equation of the perpendicular bisector of AB is the vertical line x=(c+f)/2.<br>
The circumcenter of the triangle, the intersection of those two perpendicular bisectors, is the point ((c+f)/2,(d+e)/2).<br>
But the point ((c+f)/2,(d+e)/2) is the midpoint of side AC of the triangle.<br>
So the circumcenter of the triangle is a point on the triangle.<br>
QED<br>