Prove that a cyclic quadrilateral is a rectangle
Any parallelogram cannot be cyclic. For any quadrilateral to be cyclic the sum of the opposite angles should be 180 deg. So the quadrilaterals that can be cyclic are definitely a square and a rectangle.
Prove that a cyclic quadrilateral is a rectangle
Proof that a Cyclic Quadrilateral is a Rectangle
To prove that a cyclic quadrilateral is a rectangle, we need to show that all four angles of the quadrilateral are right angles.
Let's consider a cyclic quadrilateral ABCD, where the vertices A, B, C, and D lie on a circle.
1. Properties of a Cyclic Quadrilateral
Before we begin the proof, let's understand some properties of a cyclic quadrilateral:
- Opposite angles of a cyclic quadrilateral are supplementary. This means that the sum of the opposite angles is 180 degrees.
- The opposite angles of a cyclic quadrilateral are inscribed angles that intercept the same arc on the circle.
- The angle formed by a tangent and a chord is equal to the angle subtended by the chord in the opposite segment.
- The opposite sides of a cyclic quadrilateral are equal in length.
2. Proof
Now, let's proceed with the proof:
Step 1: Let ∠BAD and ∠BCD be two opposite angles of the quadrilateral ABCD.
Step 2: Since ABCD is a cyclic quadrilateral, ∠BAD and ∠BCD are inscribed angles that intercept the same arc AD on the circle.
Step 3: By the property of inscribed angles, ∠BAD = ∠BCD.
Step 4: Since opposite angles of a cyclic quadrilateral are equal, ∠BAD = ∠BCD = x (say).
Step 5: Sum of the angles of a quadrilateral is 360 degrees. In ABCD, we have:
∠BAD + ∠ABC + ∠BCD + ∠CDA = 360 degrees.
Step 6: Substituting the value of ∠BAD and ∠BCD as x, we get:
x + ∠ABC + x + ∠CDA = 360 degrees.
Step 7: Simplifying the equation, we have:
2x + ∠ABC + ∠CDA = 360 degrees.
Step 8: Since opposite angles of a cyclic quadrilateral are supplementary, ∠ABC + ∠CDA = 180 degrees.
Step 9: Substituting the value of ∠ABC + ∠CDA as 180 degrees in the equation from Step 7, we get:
2x + 180 degrees = 360 degrees.
Step 10: Simplifying the equation, we have:
2x = 180 degrees.
Step 11: Dividing both sides of the equation by 2, we get:
x = 90 degrees.
Step 12: Therefore, both ∠BAD and ∠BCD are right angles.
Step 13: Since all four angles of the quadrilateral ABCD are right angles, it can be concluded that ABCD is a rectangle.
Hence, we have proved that a cyclic quadrilateral is a rectangle.
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