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ABCD adjacent angles angle ABC angle ACB angle BAC angle DEF angle EDF angles equal Axiom base BC BC is equal bisected centre circle ABC circumference Constr Construction Definition 15 demonstration double draw equal angles equal to F equiangular equilateral and equiangular equimultiples Euclid Euclid's Elements exterior angle fourth given circle given straight line gnomon greater ratio Hypothesis less Let ABC Let the straight meet multiple opposite angle parallel to BC parallelogram perpendicular point F polygon produced proportionals PROPOSITION 13 q.e.d. PROPOSITION rectangle contained remaining angle right angles segment shown sides similar and similarly Simson square described square on AC straight line &c THEOREM touches the circle triangle ABC triangle DEF twice the rectangle Wherefore