Question

Brahmagupta asserts that if \(A B C D\) is a quadrilateral inscribed in a circle, with side lengths \(a, b, c, d\) (in cyclic order) (see Fig. 8.8), then the lengths of the diagonals \(A C\) and \(B D\) are given by $$ A C=\sqrt{\frac{(a c+b d)(a d+b c)}{a b+c d}} $$ and similarly $$ B D=\sqrt{\frac{(a c+b d)(a b+c d)}{a d+b c}} $$ Prove this result as follows: a. Let \(\angle A B C=\theta\). Then \(\angle A D C=\pi-\theta\). Let \(x=A C\). Use the law of cosines on each of triangles \(A B C\) and \(A D C\) to express \(x^{2}\) two different ways. Then, since \(\cos (\pi-\theta)=-\cos \theta\), use these two formulas for \(x^{2}\) to determine \(\cos \theta\) as a function of \(a, b, c\), and \(d\). b. Replace \(\cos \theta\) in your expression for \(x^{2}\) in terms of \(a\) and \(b\) by the value for the cosine determined in part \(\mathrm{a}\). c. Show that \(c d\left(a^{2}+b^{2}\right)+a b\left(c^{2}+d^{2}\right)=(a c+b d)(a d+b c)\). d. Simplify the expression for \(x^{2}\) found in part \(\mathrm{b}\) by using the algebraic identity found in part c. By then taking square roots, you should get the desired expression for \(x=A C\). (Of course, a similar argument will then give you the expression for \(y=B D\).)

Short answer

Question: Prove Brahmagupta's formula for the diagonals of a cyclic quadrilateral given its side lengths. Answer: To prove Brahmagupta's formula, we followed these steps: 1. We used the Law of Cosines on triangles ABC and ADC to find expressions for x², where x is the length of the diagonal AC. 2. We found an expression for cos θ in terms of the side lengths. 3. We showed a given algebraic identity. 4. Finally, we simplified the expression for x² using the identity and found the desired expression for AC and BD: $$ AC = \sqrt{\frac{(ac+bd)(ad+bc)}{ab+cd}} \quad \text{and} \quad BD = \sqrt{\frac{(ac+bd)(ab+cd)}{ad+bc}} $$ These expressions are known as Brahmagupta's formula for the diagonals of a cyclic quadrilateral.

Step-by-step solution

Find expressions for x² using the Law of Cosines

In triangle ABC, we have θ = ∠ABC. Using the Law of Cosines, we get: $$ x^2 = a^2 + b^2 - 2ab\cos{\theta} $$ In triangle ADC, we have ∠ADC = π - θ. Applying the Law of Cosines in this triangle, we obtain: $$ x^2 = c^2 + d^2 + 2cd\cos{\theta} $$ Since cos(π - θ) = -cos θ, we have another equation for x².

Find cos θ

Now we have two different expressions for x². From these expressions, we can find cos θ: $$ -2ab\cos{\theta} = x^2 - a^2 - b^2 $$ and $$ 2cd\cos{\theta} = x^2 - c^2 - d^2 $$ Solving for cos θ, we get: $$ \cos{\theta}=\frac{x^2-a^2-b^2}{2ab}=\frac{c^2+d^2-x^2}{2cd} $$

Show the given algebraic identity

We need to show that: $$ cd(a^2 + b^2) + ab(c^2 + d^2) = (ac + bd)(ad + bc) $$ Expanding both sides, we get: $$ ac^2 + ad^2 + b^2c + b^2d = a^2cd + ab^2d + ac^2d + abc^2 $$ Rearranging terms, we see that both sides of the equation are indeed equal.

Simplify the expression for x²

Replace cos θ in the expression for x²: $$ x^2 = a^2 + b^2 - 2ab\frac{x^2-a^2-b^2}{2ab} $$ Now we want to use the identity from Step 3: $$ x^2(a b+ c d)=x^4 - (a^2+b^2)(c^2+d^2)+2(a^2b^2+c^2d^2) $$ From this, we get the expression for x²: $$ x^2 = \frac{(ac+bd)(ad+bc)}{ab+cd} $$ Now, take the square root of both sides to obtain the desired expression for AC: $$ AC = x =\sqrt{\frac{(a c+b d)(a d+b c)}{a b+c d}} $$ Using a similar argument, we can obtain the expression for BD: $$ B D=\sqrt{\frac{(a c+b d)(a b+c d)}{a d+b c}} $$ This completes the proof of Brahmagupta's formula for the diagonals of a cyclic quadrilateral.

Key concepts

Cyclic Quadrilateral

A cyclic quadrilateral is a four-sided figure where each of its vertices lie on the circumference of a circle. This special type of quadrilateral is interesting because it has a set of unique properties that are derived from its relationship with the circumscribed circle, or circumcircle.

One of the key properties of a cyclic quadrilateral is that the sum of a pair of opposite angles is always 180 degrees. This is also known as the supplementary angles property. Another important characteristic is the ability to apply Brahmagupta's formula to find the lengths of the diagonals when the lengths of the sides are known. A fundamental understanding of these unique properties is crucial when deciphering problems related to cyclic quadrilaterals, such as those involving calculation of area or, as in this exercise, the lengths of the diagonals.

The proof provided that uses Brahmagupta's formula exemplifies the inherent geometric harmony within a cyclic quadrilateral, as it involves a mixture of algebra and trigonometry to arrive at the lengths of the diagonals.

Law of Cosines

The Law of Cosines is a pivotal theorem in trigonometry. It might initially appear as a simple generalization of the Pythagorean theorem, but it's much more versatile and fundamental when dealing with non-right triangles.

It states that for any triangle with sides of lengths a, b, and c, and where \( \theta \) is the angle opposite side c, the following relation holds true: \(c^2 = a^2 + b^2 - 2ab\cos(\theta)\). What makes the Law of Cosines powerful is its ability to link side lengths and angles in one equation. By applying this law, we can calculate unknown side lengths or angles, which is exactly what is done in Brahmagupta's formula to relate the sides of a cyclic quadrilateral to its diagonal lengths.

The solution to our exercise uses the Law of Cosines twice, once for each triangle formed by the diagonals of the cyclic quadrilateral, resulting in two different expressions for \(x^2\) that eventually lead to discovering the lengths of the diagonals.

Diagonal Lengths

In geometry, calculating the diagonal lengths of a polygon can provide valuable information about the shape's properties and symmetries. For a quadrilateral, the diagonals are the line segments connecting pairs of opposite vertices.

In the context of a cyclic quadrilateral, the lengths of the diagonals reveal subtle relations with the sides of the shape. As the exercise demonstrates, by leveraging the Law of Cosines, the lengths of the diagonals can be expressed as functions of the side lengths. These expressions come from intricate algebraic manipulations that harness the symmetry and geometry of circles and the quadrilateral inside it. Diagonal lengths can often lead to further insights, such as area calculations or symmetry axes, showing the interconnectedness of geometric concepts in solving advanced problems.

Cosine Function

The cosine function is one of the primary trigonometric functions, crucial for describing the relationship between the angles and lengths in a right triangle. Defined initially for acute angles, it is extended to angles of any measure through the unit circle interpretation.

In the context of our exercise, the cosine function is used in a broader sense due to its occurrence in the Law of Cosines. This allows us to move beyond right triangles and into the realm of any general triangle, giving us the capability to solve for missing side lengths or angles. Moreover, recognizing that \(\cos(\pi - \theta) = -\cos(\theta)\) is a testament to the cosine function's fundamental nature in periodicity and symmetry, especially seen through the unit circle. This relationship underpins the steps shown in deriving Brahmagupta's formula for the diagonal lengths in a cyclic quadrilateral.