Diagonal 2 of Cyclic Quadrilateral using Ptolemy's Second Theorem Calculator

Ptolemy's Second Theorem Formula:

\[ d_2 = \frac{(S_a \times S_b) + (S_c \times S_d)}{(S_a \times S_d) + (S_b \times S_c)} \times d_1 \]

Side A of Cyclic Quadrilateral:

Side B of Cyclic Quadrilateral:

Side C of Cyclic Quadrilateral:

Side D of Cyclic Quadrilateral:

Diagonal 1 of Cyclic Quadrilateral:

Diagonal 2 of Cyclic Quadrilateral:

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1. What is Ptolemy's Second Theorem?

Ptolemy's Second Theorem relates the sides and diagonals of a cyclic quadrilateral. For a quadrilateral inscribed in a circle, the theorem provides a relationship between the lengths of its sides and diagonals, allowing calculation of one diagonal when the other diagonal and all four sides are known.

2. How Does the Calculator Work?

The calculator uses Ptolemy's Second Theorem formula:

\[ d_2 = \frac{(S_a \times S_b) + (S_c \times S_d)}{(S_a \times S_d) + (S_b \times S_c)} \times d_1 \]

Where:

\( d_2 \) — Diagonal 2 of Cyclic Quadrilateral (m)
\( S_a \) — Side A of Cyclic Quadrilateral (m)
\( S_b \) — Side B of Cyclic Quadrilateral (m)
\( S_c \) — Side C of Cyclic Quadrilateral (m)
\( S_d \) — Side D of Cyclic Quadrilateral (m)
\( d_1 \) — Diagonal 1 of Cyclic Quadrilateral (m)

Explanation: The formula calculates the second diagonal based on the product relationships between the sides and the known first diagonal.

3. Importance of Diagonal Calculation

Details: Calculating diagonals in cyclic quadrilaterals is crucial for geometric analysis, architectural design, and engineering applications where circular or cyclic properties are involved.

4. Using the Calculator

Tips: Enter all four side lengths and the first diagonal length in meters. All values must be positive numbers greater than zero.

5. Frequently Asked Questions (FAQ)

Q1: What is a cyclic quadrilateral?
A: A cyclic quadrilateral is a four-sided polygon where all vertices lie on a single circle.

Q2: When does the formula return undefined?
A: The formula becomes undefined when the denominator \((S_a \times S_d) + (S_b \times S_c)\) equals zero.

Q3: Can this theorem be applied to any quadrilateral?
A: No, Ptolemy's theorems only apply to cyclic quadrilaterals (those that can be inscribed in a circle).

Q4: What are practical applications of this calculation?
A: This calculation is used in geometry problems, architectural design of circular structures, and engineering applications involving cyclic configurations.

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect cyclic quadrilaterals, assuming precise input measurements.