In a line segment AB point C is called a mid-point of line segment AB. Prove that every line segment has one and only one midpoint.

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In a line segment AB point C is called a mid-point of line segment AB. Prove that every line segment has one and only one midpoint.

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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take c(0,0) on origin and take co-ordinates of point A(-a,0) and of point B(a,0) By applying distance formula prove that AC = CB...and hence proved, if you want complete solution I can provide it to you.
I have to prove it using Euclid's Axioms.
A line is represented by a linear equation. So when you solve this equation for midpoint you will also get a linear equation. And you know A linear equation has only one solution. So you will get only one value for midpoint.

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To prove, you need show 2 different midpoints are the same. Let \(A=(X_A,Y_A), B=(X_B,Y_B)\) Suppose C, C' both are midpoints of AB. Since C is midpoint of AB, the coordinate of C is \(C =(\dfrac{X_A+X_B}{2}, \dfrac{Y_A+Y_B}{2})\) Since C' is also midpoint of AB, the coordinate of C' is \(C'=(\dfrac{X_A+X_B}{2},\dfrac{Y_A+Y_B}{2})\) Hence C = C'\(\implies\) there is only 1 midpoint of the segment AB

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