Diagonal 1 of Cyclic Quadrilateral using Ptolemy's Second Theorem Calculator

Ptolemy's Second Theorem:

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

Side A of Cyclic Quadrilateral:

Side B of Cyclic Quadrilateral:

Side C of Cyclic Quadrilateral:

Side D of Cyclic Quadrilateral:

Diagonal 2 of Cyclic Quadrilateral:

Diagonal 1 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. It states that for a cyclic quadrilateral, the ratio of the diagonals can be expressed in terms of the sides of the quadrilateral.

2. How Does the Calculator Work?

The calculator uses Ptolemy's Second Theorem:

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

Where:

\( d_1 \) — Diagonal 1 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)
\( d_2 \) — Diagonal 2 of Cyclic Quadrilateral (m)

Explanation: The theorem provides a relationship between the diagonals and sides of a cyclic quadrilateral, allowing calculation of one diagonal when the other diagonal and all four sides are known.

3. Importance of Diagonal Calculation

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

4. Using the Calculator

Tips: Enter all four side lengths and the second diagonal in meters. All values must be positive numbers. The calculator will compute the first diagonal using Ptolemy's Second Theorem.

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 denominator become zero?
A: The denominator becomes zero when (S_a × S_b) + (S_c × S_d) = 0, which is not possible for positive side lengths.

Q3: Can this theorem be used for any quadrilateral?
A: No, Ptolemy's theorems apply specifically to cyclic quadrilaterals where all vertices lie on a circle.

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

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for perfect cyclic quadrilaterals, following Ptolemy's proven theorem.