If the quadrilateral has an inscribed circle, then the sum of a pair of opposite sides equals:
Sum of other pair of opposite sides
Understanding Quadrilaterals with Inscribed Circles
A quadrilateral that has an inscribed circle is called a tangential quadrilateral. This special type of quadrilateral has a circle inside it that is tangent to all four of its sides. There is a significant property that holds true for all tangential quadrilaterals, relating the lengths of their sides.
Key Property of Tangent Segments
A fundamental geometric property states that if two tangent segments are drawn to a circle from the same external point, then these segments have equal lengths.
Consider a tangential quadrilateral ABCD, with an inscribed circle. Let the points where the circle touches the sides AB, BC, CD, and DA be P, Q, R, and S respectively.
Applying the tangent segment property at each vertex:
- From vertex A: The tangent segments are AP and AS. So, \( \text{AP} = \text{AS} \).
- From vertex B: The tangent segments are BP and BQ. So, \( \text{BP} = \text{BQ} \).
- From vertex C: The tangent segments are CQ and CR. So, \( \text{CQ} = \text{CR} \).
- From vertex D: The tangent segments are DR and DS. So, \( \text{DR} = \text{DS} \).
Applying the Tangent Property to Sides
The length of each side of the quadrilateral is the sum of the lengths of the two tangent segments that make up that side:
- Side AB = AP + PB
- Side BC = BQ + QC
- Side CD = CR + RD
- Side DA = DS + SA
Using the equality of tangent segments from each vertex, we can rewrite the side lengths:
| Side | Expressed using tangent segments | Using tangent equality |
|---|---|---|
| AB | AP + PB | AS + BQ |
| BC | BQ + QC | BQ + CR |
| CD | CR + RD | CR + DS |
| DA | DS + SA | DS + AP |
Deriving the Relationship Between Opposite Sides
Now let's consider the sum of the lengths of opposite sides in the quadrilateral ABCD.
Consider the sum of one pair of opposite sides, say AB and CD:
\( \text{AB} + \text{CD} = (\text{AP} + \text{PB}) + (\text{CR} + \text{RD}) \)
Using the tangent segment equalities (AP = AS, PB = BQ, CR = CQ, RD = DS):
\( \text{AB} + \text{CD} = (\text{AS} + \text{BQ}) + (\text{CQ} + \text{DS}) \)
\( \text{AB} + \text{CD} = \text{AS} + \text{BQ} + \text{CQ} + \text{DS} \)
Now consider the sum of the other pair of opposite sides, say BC and DA:
\( \text{BC} + \text{DA} = (\text{BQ} + \text{QC}) + (\text{DS} + \text{SA}) \)
Using the tangent segment equalities (QC = CR, SA = AP):
\( \text{BC} + \text{DA} = (\text{BQ} + \text{CR}) + (\text{DS} + \text{AP}) \)
\( \text{BC} + \text{DA} = \text{BQ} + \text{CR} + \text{DS} + \text{AP} \)
Comparing the sums:
\( \text{AB} + \text{CD} = \text{AS} + \text{BQ} + \text{CQ} + \text{DS} \)
\( \text{BC} + \text{DA} = \text{AP} + \text{BQ} + \text{CR} + \text{DS} \)
Since AP = AS and CQ = CR, we can see that the set of tangent segment lengths {AS, BQ, CQ, DS} is the same as {AP, BQ, CR, DS}.
Therefore, \( \text{AB} + \text{CD} = \text{BC} + \text{DA} \).
Conclusion on Opposite Sides Sum
For any quadrilateral with an inscribed circle (a tangential quadrilateral), the sum of the lengths of one pair of opposite sides is equal to the sum of the lengths of the other pair of opposite sides. This property is known as Pitot's theorem for quadrilaterals.
Let the sides be a, b, c, d in order around the quadrilateral. Pitot's theorem states that if an inscribed circle exists, then \( a + c = b + d \).
Reviewing the options:
- Half the sum of the diagonals
- Sum of other pair of opposite sides
- Sum of two adjacent sides
- None of the above
Our derivation shows that the sum of a pair of opposite sides equals the sum of the other pair of opposite sides. This matches option 2.
Revision Table: Tangential Quadrilateral Properties
| Property | Description |
|---|---|
| Inscribed Circle Existence | A circle is tangent to all four sides internally. |
| Tangent Segment Property | Tangents from a vertex to the circle are equal in length. |
| Pitot's Theorem | Sum of one pair of opposite sides equals the sum of the other pair of opposite sides (\(a+c = b+d\)). |
Additional Information: Related Concepts
The concept of a quadrilateral having an inscribed circle is a special case within geometry. Here are some related ideas:
- Tangential Polygons: Pitot's theorem can be generalized to any tangential polygon with an even number of sides. The alternating sum of side lengths is zero. For an odd number of sides, this doesn't directly apply in the same way.
- Circumscribed Circle: A quadrilateral that has a circumscribed circle (a circle passing through all four vertices) is called a cyclic quadrilateral. Cyclic quadrilaterals have different properties, such as the sum of opposite angles being 180 degrees.
- Bicentric Quadrilateral: A quadrilateral that has both an inscribed circle and a circumscribed circle is called a bicentric quadrilateral. These quadrilaterals possess properties of both tangential and cyclic quadrilaterals.
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