Prove that a line has only one midpoint using Euclid's axioms.

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Solution

Let there be 2 midpoints C and D of line AB

When C is MP, AC =BC ---- eqn 1

When D is MP, AD = BD ---- eqn2

Subtract eqn 1 frm 2

AD - AC = BD - BC

CD= -CD (writing DC as -CD means the same)

CD + CD = 0

2CD = 0

CD = 0

This is possible only when C coincides with D

Hence there is only 1 midpoint.

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Prove that a line has only one midpoint using Euclid's axioms.

Let there be 2 midpoints C and D of line AB When C is MP, AC =BC ---- eqn 1 When D is MP, AD = BD ---- eqn2 Subtract eqn 1 frm 2 AD - AC = BD - BC CD= -CD (writing DC as -CD means the same) CD + CD = 0 2CD = 0 CD = 0 This is possible only when C coincides with D Hence there is only 1 midpoint.