A semicircle is the figure contained by the diameter and the circumference cut off by it. And the center of the semicircle is the same as that of the circle.
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"The properties of figures are derived from their definitions and postulates."
"To cut off from the greater of two given unequal straight lines a straight line equal to the less."
"The greatest of the parts is called the antecedent, and the less the consequent."
"A figure is that which is contained by any boundary or boundaries."
"An acute angle is an angle less than a right angle."
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A semicircle is defined by two components: the straight diameter line and the curved arc it cuts from a circle. Crucially, the center point of the semicircle is identical to the center of the original circle. This establishes a precise, unambiguous definition where geometric relationships are fixed and derivable from first principles, leaving no room for vagueness or interpretation.
Euclid built geometry on rigorous definitions before proving anything, as seen in his Elements. This passage exemplifies his method: establish exact terms first, then build theorems upon them. His career at Alexandria's Library involved systematizing scattered Greek mathematical knowledge into an airtight logical framework where every concept rested on clearly stated foundations.
Around 300 BCE, Greek civilization was transitioning from empirical measurement traditions inherited from Egypt and Babylon toward abstract, proof-based mathematics. Alexandria under the Ptolemies became the intellectual hub of the Mediterranean world. Establishing definitions this precisely was revolutionary, replacing practical rule-of-thumb geometry with a deductive science capable of universal truths independent of physical measurement.
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