A,B,C,D cocyclic prove AC^2=AB×AD+BC^2

MHB
Thread starter Albert1
Start date Apr 24, 2017

In summary, cocyclic is a term used in geometry to describe a set of points that lie on the same circle, forming a cyclic quadrilateral. The theorem of intersecting chords can be used to prove that AC^2=AB×AD+BC^2 for cocyclic points. The steps to prove this statement involve drawing a diagram, setting up an equation, simplifying and isolating terms, and using the properties of equality. This statement can be proved for any set of cocyclic points as long as the intersecting chords theorem can be applied. The concept of cocyclic points has applications in geometry, physics, and engineering.

Apr 24, 2017

Albert1

$ABCD$ is a Cyclic quadrilateral,given $BC=CD$

Prove:

$AC^2=AB\times AD+BC^2$

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Apr 28, 2017

Albert1

Albert said:

$ABCD$ is a Cyclic quadrilateral,given $BC=CD$

Prove:

$AC^2=AB\times AD+BC^2$

hint:

using the following diagram:
View attachment 6569

1. What is the concept of cocyclic in geometry?

Cocyclic is a term used in geometry to describe a set of points that lie on the same circle. In other words, these points can be connected to form a cyclic quadrilateral (a four-sided figure with all its vertices lying on a common circle).

2. How do you prove that AC^2=AB×AD+BC^2 for cocyclic points?

In order to prove that AC^2=AB×AD+BC^2 for cocyclic points, you can use the theorem of the intersecting chords. This theorem states that if two chords intersect inside a circle, then the product of the segments of one chord is equal to the product of the segments of the other chord. In this case, AC and BD are the chords intersecting at point C, and AB, AD, and BC are the segments of those chords.

3. What are the steps to prove that AC^2=AB×AD+BC^2?

The steps to prove this statement would be:
1. Draw a diagram of the given cocyclic points and label all known values.
2. Use the theorem of the intersecting chords to set up an equation using the given values.
3. Simplify the equation by substituting in the known values.
4. Use algebraic manipulation to isolate AC^2 on one side of the equation.
5. Simplify the other side of the equation to match the isolated AC^2 term.
6. Use the properties of equality to show that both sides of the equation are equal, thus proving the statement.

4. Can the statement AC^2=AB×AD+BC^2 be proved for any set of cocyclic points?

Yes, the statement AC^2=AB×AD+BC^2 can be proved for any set of cocyclic points as long as the intersecting chords theorem can be applied to those points. As long as the points form a cyclic quadrilateral, the theorem can be used to prove this statement.

5. How is the concept of cocyclic points useful in geometry and other fields of science?

The concept of cocyclic points is useful in geometry as it allows us to prove various theorems and statements, such as the intersecting chords theorem. It also has applications in other fields of science, such as physics and engineering, where the concept of circles and cyclic patterns is commonly used to understand and analyze various phenomena.