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This discussion aims to illuminate the mathematical properties of **secant circles**, their relationship with **tangents**, and the rich tapestry of applications they find in **science**, **engineering**, and **everyday life**.

**Definition of Secant Circle**

In geometry, a **secant circle**, with respect to another circle, is a circle that intersects the given circle at exactly two distinct points. These two points are the **secant points**. If the circles intersect at more than two points or touch each other at one point (becoming **tangent circles**), they are not considered secant circles.

A **“secant circle”** isn’t a standard term in **geometry**. However, “secant” in geometry usually refers to a **line intersecting a circle at exactly two points**. When applied to two circles, it could mean that the circles intersect at exactly two points.

Hence, **“secant circles”** can be interpreted as two circles **intersecting** at exactly two points. In this context, the **common chord** that connects the two points of **intersection** can be considered a **“secant”** to both** circles**.

Figure-1.

**Properties ****of Secant Circle**

While **secant circles** do not have distinctive properties, they are part of the wider study of **circle geometry** involving intersections, secants, and tangents. A secant circle, by definition, **intersects another circle at two distinct points**. Here are the relevant properties that could apply to secant circles:

**Intersection Points**

A **secant circle** intersects the given circle at exactly **two distinct points**. If there is only **one point of intersection**, the circles are considered **tangent**, not secant.

**Common Chord**

The **line segment** joining the two points of intersection forms a **common chord** for the two intersecting circles. This chord divides each circle into two arcs: a **major arc** and a **minor arc**.

**Angle Between Radii**

The **angle** between the radii drawn from the center of a circle to the two intersection points is called the **intercepted angle**. The measure of this angle is equal to **half the difference** of the measures of the **intercepted arcs**.

**Power of a Point**

The **product** of the lengths of two line segments from a point **outside** a circle to the points of intersection with the circle (when drawn as secant lines) is **constant** for all such points on a **single line** passing through the point and the center of the circle.

**Cross-Ratio**

If two** circles** are **secant** at points** A** and **B,** any line through **A** and **B** **intersects** the circles again at points **C** and **D** (in some order), and the **cross-ratio** of points **A, B, C, D** is real and the same for all such l**ines**. This** cross-ratio** is called the **radical ratio** of the two circles.

**Radical Axis**

The **locus of points** having equal power with respect to two circles forms a line known as the **radical axis**. The radical axis of two circles is **perpendicular to the line joining the centers** of the circles and **passes through the midpoint** of the segment joining the two intersection points.

**Orthogonality**

Two circles are said to be **orthogonal** if their **radii at the points of intersection are perpendicular**. For two **secant circles** to be orthogonal, the **square of the distance between their centers equals the sum of the squares of their radii**.

**Coaxial System of Circles**

If a system of **circles** all **intersect** a given circle **orthogonally** (i.e., at right angles), they are said to form a coaxial system of circles, sharing a common **radical axis**.

It’s worth noting that the** relationships** and properties mentioned are derived from the principles of **Euclidean geometry** and can be applied in solving various** geometric problems** involving **circles** and their **intersections**.

**Ralevent Formulas **

**Secant circles** don’t have a **unique** set of **formulas** because they are defined merely as two circles **intersecting** at exactly two points. However, related geometrical concepts, like **intersecting circles** and **chord lengths**, have several important **formulas**:

**Length of the Common Chord**

If two circles intersect and you know the **radii** of both circles (**r1** and **r2**) and the **distance** between their **centers** (**d**), you can calculate the length of the **common chord** (**l**) using the **formula**:

l = 2 * √[r1 * (r1 – d)] if d < r1 < 2d (if the center of the circle with radius r1 lies outside the other circle)

l = 2 * √[r2 * (2d – r2)] if d < r2 < 2d (if the center of the circle with radius r2 lies outside the other circle)

**Radical Axis**

The **radical axis** of two circles is a line that passes through the two **points of intersection** of the circles. If the **equations** of the circles are given by:

(x – a1)² + (y – b1)² = r1²

and

(x – a2)² + (y – b2)² = r2²

then the equation of the radical axis is given by:

(x – a1)² + (y – b1)² – r1² = (x – a2)² + (y – b2)² – r2²

**Radical Center**

If you have three circles **pairwise secant** (each pair of circles intersects at two distinct points), they have a common **radical axis** that passes through a point known as the **radical center**. The **coordinates** of the radical center are found by equating the pairwise radical axes and solving for **(x, y)**.

**Power of a Point**

If a point is outside a **circle** and is connected to the circle by two lines forming a **chord**, the **power** of the point with respect to the circle is given by the **product** of the **lengths** of the two **line segments**, which is a constant for all **chords** through that point. If the lengths are denoted by **d1** and **d2**, the power is given by **d1*d2**.

**Exercise **

**Example 1**

Two **secant circles** have radii of **5 cm** and **13 cm**, and their centers are **12 cm** apart. What is the length of the **common chord** of the two circles?

Figure-2.

**Solution**

The length of the common chord is:

`l = 2*(√(13² - 6²) - √(5² - 6²)) `

`l = 2*(√(145) - √(7)) `

l = 16.528 `cm`

**Example 2**

Two circles with radii** 7 cm** and **9 cm** are **secant**, with their centers **10 cm** apart. Determine the length of the **common chord**.

**Solution**

The length of the common chord is:

`l = 2*(√(9² - 5²) + √(7² - 5²)) `

`l = 2*(√(14) + √(6)) `

l = 12.198`cm`

**Example 3**

Two circles with radii **4 cm** and **6 cm** are **secant**, and their centers are **8 cm** apart. Find the length of the com**mon chord**.

Figure-3.

**Solution**

The length of the common chord can be calculated using the formula; ** 2*√(r² - (d/2)²)**, where

**is the circle’s radius and the distance between the centers of the two circles.**

`r`

Therefore, the length of the common chord is:

ls =`2*√(4² - (8/2)²) `

`ls = = 2*√(16 - 16) `

`ls = 0 cm`

for the smaller circle. And:

`ll = 2*√(6² - (8/2)²) `

`ll = 2*√(36 - 16) `

`ll = 2*√(20) cm`

for the larger circle. Thus, the length of the common chord is ** 2*√(20) cm**.

**Example 4**

Two** secant circles** intersect at points **A** and **B.** The circles have radii of **15 cm** and **9 cm**, and the distance between their centers is **14 cm**. What is the length of **AB**?

**Solution**

The length of AB can be calculated as:

l = √`(15² - 7²) + √(9² - 7²) `

`l = √(134) + √(16) `

`l = √(134) + 4 `

`l = 14.032 cm`

**Example 5**

Two **secant circles** have **8 cm** and **12 cm** radii, and their centers are **16 cm** apart. What is the length of the **common chord** of the two circles?

**Solution**

The length of the common chord can be calculated as follows:

`l = 2*√(12² - (16/2)²) `

`l = 2*√(144 - 64) `

`l = 2*√(80) cm`

for the larger and smaller circles. Thus, the length of the** common chord** is ** 2*√(80) cm**.

**Applications **

While **secant circles** might not directly find many applications in various fields, the underlying principles of **intersecting circles** and **lines** play a vital role in multiple disciplines. Some areas where these concepts come into play include:

**Physics**

In **physics**, especially **optics**, the concept of **secant circles** is utilized to describe phenomena like **lens distortion**, **spherical aberrations**, and the formation of **rainbows**.

**Engineering**

Civil engineering principles underlying **secant circles** help design **roundabouts**, **curved paths**, and **architectural planning**. In **mechanical engineering**, **gears** are designed using **intersecting circles**, which, in a way, behave like **secant circles**.

**Astronomy**

The **geometry** of circles is employed to measure the distances between **celestial bodies**, to analyze the trajectories of **planets** and **satellites**, and to calculate the **apparent sizes** of stars and planets.

**Computer Graphics and Image Processing**

Algorithms for drawing **circles** or **arcs** and detecting circular objects in images often involve principles associated with **secant circles**.

**Geography and Cartography**

Concepts of circle geometry, including **tangent** and **secant properties**, are utilized in **map projections**, where the Earth’s spherical shape is translated onto a two-dimensional surface.

**Navigation**

In the **aviation** and **maritime** sectors, the geometry of the Earth (treated as a sphere or an oblate spheroid) is understood in terms of **great circles** and **small circles**, which are related to the concept of **secant circles**.

Remember that while the concept of **secant circles** might not be directly applied, the **mathematical principles** and the underlying **geometric intuition** form the** foundation** for many applications in these fields.

*All images were created with GeoGebra.*