Euclid Element Book 1 proposition 3

1. Proposition - 1
Raman Choubay

2. Proposition 1• To construct an equilateral triangle on a given finite
straight-line.
• Step for construction
Describe the circle BCD with centre A and radius AB.
Again describe the circle ACE with centre B and radius BA.
Join the straight lines CA and CB from the point C at which the circles cut one another to
the points A and B

3. Proof
Statements Reason
AB = AC ∵ 𝐴 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒𝐶𝐷𝐵
BA = BC ∵ 𝐵 𝑖𝑠 𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝐶𝐴𝐸
⇒ 𝐴𝐶 = 𝐶𝐵 ∵ 𝐴𝐵 = 𝐴𝐶 𝑎𝑛𝑑 𝑎𝑙𝑠𝑜 𝐴𝐵 = 𝐵𝐶 𝑎𝑛𝑑
Things equal to the same thing are
also equal to one another
AB = AC = BC ∵ 𝐴𝐵 = 𝐴𝐶, 𝐵𝐴 = 𝐵𝐶 𝐴𝑁𝐷 𝐴𝐶 = 𝐶𝐵
Thus, the triangle ABC is equilateral, and has been constructed on the given finite
straight-line AB.
Q.E.F