Proposition 22
Theorem
The sum of the opposite angles of a quadrilateral inscribed in a circle is two right angles.
Commentary
1. Let ◇ABCD be a quadrilateral inscribed in a circle.
2. Then opposite angles of◇ABCD sum to two right angles or 180°. Namely, ∠ABC + ∠ADC = 180 ° and ∠BAD + ∠BCD = 180 °.
3. It follows that the sum of the interior angles of a quadrilateral inscribed in a circle is 360°.
4. Book 1 Proposition 32 discusses the sum of the interior angles of a triangle.
2. Then opposite angles of
3. It follows that the sum of the interior angles of a quadrilateral inscribed in a circle is 360°.
4. Book 1 Proposition 32 discusses the sum of the interior angles of a triangle.
Original statement
τῶν ἐν τοῖς κύκλοις τϵτραπλϵύρων αἱ ἀπϵναντίον γωνίαι δυσὶν ὀρθαῖς ἴσαι ϵἰσίν.
English translation
The opposite angles of quadrilaterals in circles are equal to two right angles.
