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Proof of Euclid's Proposition 47
In right-angled triangles the square on
the side opposite the right angle equals the sum of the squares on the sides
containing the right angle.
- Construct three squares with AB, BC, and AC being one side of each square.
Proposition 46, Proposition31, Postulate 1
- Since angles BAC and BAG are right angles, it follows by Postulate 14 that the CA is
a straight line with AG.
- By the same argument, BA is a straight line with AH.
- Since the angles FBA and DBC are right angles, they are equal to each other.
- Since the angle FBC is the sum of angles FBA and ABC, and angle ABD is the sum of the
angles ABC and DBC, and FBA is equal to DBC, we can conclude that angle FBC is equal to
angle ABC.
- Since the length of FB equals the length of AB by the definition of a square, and the
length of BC equals the length of BD by the definition of a square, we can conclude by
Proposition 4 (SAS) that triangle FBC is congruent to triangle ABD.
- By a similar argument, triangle KCB is congruent with triangle ACE.
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