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URL: https://nakafa.com/en/subjects/mathematics/circle/external-tangent-line-and-internal-tangent-line
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/circle/external-tangent-line-and-internal-tangent-line/en.mdx

Learn external and internal tangent lines between circles through formulas, calculations, and worked examples.

---

## Definition of Common Tangent Lines

A common tangent line is a line that touches two circles simultaneously. There are two types of common tangent lines:

1. **External Common Tangent Line**: A line that touches both circles from the same side
2. **Internal Common Tangent Line**: A line that touches both circles from opposite sides

## Concept of External Common Tangent Line

An external common tangent line is a line that touches both circles and does not intersect the line connecting the two circle centers.

Component: LineEquation
Props:
- title: External Common Tangent Line
- description: Two external common tangent lines on two circles.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: -4 + 2 * Math.cos(angle),
y: 2 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 + 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: -4, y: 0, z: 0 },
{ x: 4, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "B", at: 1, offset: [0.5, -0.5, 0] },
],
},
{
points: (() => {
const O1 = { x: -4, y: 0 };
const O2 = { x: 4, y: 0 };
const r1 = 2;
const r2 = 3;
const d = Math.sqrt((O2.x - O1.x) ** 2 + (O2.y - O1.y) ** 2);

// Calculate the angle for external tangent
const sin_alpha = (r2 - r1) / d;
const cos_alpha = Math.sqrt(1 - sin_alpha * sin_alpha);

// Direction from O1 to O2
const dx = O2.x - O1.x;
const dy = O2.y - O1.y;

// Normalize direction
const ux = dx / d;
const uy = dy / d;

// Perpendicular direction (rotated 90 degrees)
const vx = -uy;
const vy = ux;

// First tangent line (upper)
// Tangent point on ... [truncated; 3085 chars]
- cameraPosition: [0, 0, 15]
- showZAxis: false

### Formula for External Common Tangent Line Length

For two circles with:

- First circle center: $$O_1$$
- Second circle center: $$O_2$$
- First circle radius: $$r_1$$
- Second circle radius: $$r_2$$
- Distance between centers: $$d$$

Visible text: - First circle center: 
- Second circle center: 
- First circle radius: 
- Second circle radius: 
- Distance between centers:

Length of external common tangent line:

```math
l = \sqrt{d^2 - (r_2 - r_1)^2}
```

### Finding the Length of External Common Tangent Line

Two circles are centered at $$A(-3, 0)$$ with radius $$1.5$$ and $$B(3, 0)$$ with radius $$2.5$$. Find the length of the external common tangent line!

Visible text: Two circles are centered at with radius and with radius . Find the length of the external common tangent line!

Component: LineEquation
Props:
- title: Circles with $$r_1 = 1.5$$ and{" "}
$$r_2 = 2.5$$
  Visible text: Circles with and{" "}
- description: Visualization of external common tangent line.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: -3 + 1.5 * Math.cos(angle),
y: 1.5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 + 2.5 * Math.cos(angle),
y: 2.5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: -3, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "B", at: 1, offset: [0.5, -0.5, 0] },
],
},
{
points: (() => {
const O1 = { x: -3, y: 0 };
const O2 = { x: 3, y: 0 };
const r1 = 1.5;
const r2 = 2.5;
const d = Math.sqrt((O2.x - O1.x) ** 2 + (O2.y - O1.y) ** 2);

// Calculate the angle for external tangent
const sin_alpha = (r2 - r1) / d;
const cos_alpha = Math.sqrt(1 - sin_alpha * sin_alpha);

// Direction from O1 to O2
const dx = O2.x - O1.x;
const dy = O2.y - O1.y;

// Normalize direction
const ux = dx / d;
const uy = dy / d;

// Perpendicular direction (rotated 90 degrees)
const vx = -uy;
const vy = ux;

// First tangent line (upper)
// Tange ... [truncated; 2920 chars]
- cameraPosition: [0, 0, 12]
- showZAxis: false

**Solution:**

Component: MathContainer
Children:

```math
d = \sqrt{(3-(-3))^2 + (0-0)^2} = \sqrt{36} = 6
```

```math
l = \sqrt{d^2 - (r_2 - r_1)^2}
```

```math
l = \sqrt{6^2 - (2.5 - 1.5)^2}
```

```math
l = \sqrt{36 - 1}
```

```math
l = \sqrt{35}
```

Therefore, the length of the external common tangent line is $$\sqrt{35} \text{ units}$$.

Visible text: Therefore, the length of the external common tangent line is .

## Concept of Internal Common Tangent Line

An internal common tangent line is a line that touches both circles from opposite sides and intersects the line connecting the two circle centers.

Component: LineEquation
Props:
- title: Internal Common Tangent Line
- description: Two internal common tangent lines on two circles.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: -4 + 2 * Math.cos(angle),
y: 2 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 4 + 2 * Math.cos(angle),
y: 2 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: -4, y: 0, z: 0 },
{ x: 4, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "A", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "B", at: 1, offset: [0.5, -0.5, 0] },
],
},
{
points: (() => {
const O1 = { x: -4, y: 0 };
const O2 = { x: 4, y: 0 };
const r1 = 2;
const r2 = 2;
const d = Math.sqrt((O2.x - O1.x) ** 2 + (O2.y - O1.y) ** 2);

const alpha = Math.asin((r1 + r2) / d);
const theta = Math.atan2(O2.y - O1.y, O2.x - O1.x);

const T1x = O1.x + r1 * Math.cos(theta + Math.PI / 2 - alpha);
const T1y = O1.y + r1 * Math.sin(theta + Math.PI / 2 - alpha);

const T2x = O2.x + r2 * Math.cos(theta + Math.PI / 2 - alpha + Math.PI);
const T2y = O2.y + r2 * Math.sin(theta + Math.PI / 2 - alpha + Math.PI);

const dx = T2x - T1x;
const dy = T2 ... [truncated; 2375 chars]
- cameraPosition: [0, 0, 15]
- showZAxis: false

### Formula for Internal Common Tangent Line Length

Length of internal common tangent line:

```math
l = \sqrt{d^2 - (r_1 + r_2)^2}
```

**Condition:** Internal common tangent lines exist only if $$d > r_1 + r_2$$ (the two circles do not intersect).

Visible text: **Condition:** Internal common tangent lines exist only if (the two circles do not intersect).

### Finding the Length of Internal Common Tangent Line

Two circles are centered at $$P(-5, 0)$$ with radius $$2$$ and $$Q(5, 0)$$ with radius $$3$$. Find the length of the internal common tangent line!

Visible text: Two circles are centered at with radius and with radius . Find the length of the internal common tangent line!

Component: LineEquation
Props:
- title: Circles with $$r_1 = 2$$ and{" "}
$$r_2 = 3$$
  Visible text: Circles with and{" "}
- description: Visualization of internal common tangent line.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: -5 + 2 * Math.cos(angle),
y: 2 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 5 + 3 * Math.cos(angle),
y: 3 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: -5, y: 0, z: 0 },
{ x: 5, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "P", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "Q", at: 1, offset: [0.5, -0.5, 0] },
],
},
{
points: (() => {
const O1 = { x: -5, y: 0 };
const O2 = { x: 5, y: 0 };
const r1 = 2;
const r2 = 3;
const d = Math.sqrt((O2.x - O1.x) ** 2 + (O2.y - O1.y) ** 2);

const alpha = Math.asin((r1 + r2) / d);
const theta = Math.atan2(O2.y - O1.y, O2.x - O1.x);

const T1x = O1.x + r1 * Math.cos(theta + Math.PI / 2 - alpha);
const T1y = O1.y + r1 * Math.sin(theta + Math.PI / 2 - alpha);

const T2x = O2.x + r2 * Math.cos(theta + Math.PI / 2 - alpha + Math.PI);
const T2y = O2.y + r2 * Math.sin(theta + Math.PI / 2 - alpha + Math.PI);

return [
{ x: T1x, y: T1y, z: 0 }, ... [truncated; 2142 chars]
- cameraPosition: [0, 0, 18]
- showZAxis: false

**Solution:**

First, check if internal common tangent lines exist:

Component: MathContainer
Children:

```math
d = \sqrt{(5-(-5))^2 + (0-0)^2} = \sqrt{100} = 10
```

```math
r_1 + r_2 = 2 + 3 = 5
```

```math
d = 10 > 5 = r_1 + r_2 \space \checkmark
```

Since the condition is satisfied, then:

Component: MathContainer
Children:

```math
l = \sqrt{d^2 - (r_1 + r_2)^2}
```

```math
l = \sqrt{10^2 - 5^2}
```

```math
l = \sqrt{100 - 25}
```

```math
l = \sqrt{75}
```

```math
l = 5\sqrt{3}
```

Therefore, the length of the internal common tangent line is $$5\sqrt{3} \text{ units}$$.

Visible text: Therefore, the length of the internal common tangent line is .

## Circles with Equal Radii

When two circles have equal radii ($$r_1 = r_2 = r$$), there are special properties:

Visible text: When two circles have equal radii (), there are special properties:

### External Common Tangent Line

Component: LineEquation
Props:
- title: External Common Tangent Line for Equal Radii
- description: External common tangent lines are parallel to the line of centers.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: -3 + 2 * Math.cos(angle),
y: 2 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 3 + 2 * Math.cos(angle),
y: 2 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: -3, y: 0, z: 0 },
{ x: 3, y: 0, z: 0 },
],
color: getColor("ORANGE"),
showPoints: true,
labels: [
{ text: "O₁", at: 0, offset: [-0.5, -0.5, 0] },
{ text: "O₂", at: 1, offset: [0.5, -0.5, 0] },
],
},
{
points: [
{ x: -3, y: 2, z: 0 },
{ x: 3, y: 2, z: 0 },
],
color: getColor("CYAN"),
showPoints: false,
smooth: false,
},
{
points: [
{ x: -3, y: -2, z: 0 },
{ x: 3, y: -2, z: 0 },
],
color: getColor("TEAL"),
showPoints: false,
smooth: false,
},
]
- cameraPosition: [0, 0, 12]
- showZAxis: false

For $$r_1 = r_2$$:

Visible text: For :

- External common tangent lines are **parallel** to the line connecting the two centers
- Length of external common tangent line is $$d$$ (distance between centers)

Visible text: - External common tangent lines are **parallel** to the line connecting the two centers
- Length of external common tangent line is (distance between centers)

## Cases of Various Circle Positions

Determine the length of external and internal common tangent lines for the following circles:

### Distant Circles

The first circle is centered at $$(-6, 0)$$ with radius $$1$$, the second circle is centered at $$(6, 0)$$ with radius $$2$$.

Visible text: The first circle is centered at with radius , the second circle is centered at with radius .

Component: LineEquation
Props:
- title: Case: Distant Circles
- description: Both types of common tangent lines exist.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: -6 + 1 * Math.cos(angle),
y: 1 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 6 + 2 * Math.cos(angle),
y: 2 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: (() => {
const O1 = { x: -6, y: 0 };
const O2 = { x: 6, y: 0 };
const r1 = 1;
const r2 = 2;
const d = Math.sqrt((O2.x - O1.x) ** 2 + (O2.y - O1.y) ** 2);

// Calculate the angle for external tangent
const sin_alpha = (r2 - r1) / d;
const cos_alpha = Math.sqrt(1 - sin_alpha * sin_alpha);

// Direction from O1 to O2
const dx = O2.x - O1.x;
const dy = O2.y - O1.y;

// Normalize direction
const ux = dx / d;
const uy = dy / d;

// Perpendicular direction (rotated 90 degrees)
const vx = -uy;
const vy = ux;

// First tangent line (upper)
// Tangent point on first circle
const T1x = O1.x + r1 * (sin_alpha * ux + cos_alpha * vx);
const T1y = O1.y + r1 * (sin_alpha * uy + cos_alpha * vy);

// Tangent point on second circle
const T2x = O2.x + r2 * (sin_alpha * ux + cos_al ... [truncated; 4353 chars]
- cameraPosition: [0, 0, 18]
- showZAxis: false

**Solution:**

Component: MathContainer
Children:

```math
d = \sqrt{(6-(-6))^2 + (0-0)^2} = 12
```

```math
l_{external} = \sqrt{12^2 - (2-1)^2} = \sqrt{144 - 1} = \sqrt{143}
```

```math
l_{internal} = \sqrt{12^2 - (1+2)^2} = \sqrt{144 - 9} = \sqrt{135} = 3\sqrt{15}
```

### Close Circles

The first circle is centered at $$(-2, 0)$$ with radius $$1.5$$, the second circle is centered at $$(2, 0)$$ with radius $$1.5$$.

Visible text: The first circle is centered at with radius , the second circle is centered at with radius .

Component: LineEquation
Props:
- title: Case: Close Circles with Equal Radii
- description: External common tangent lines are parallel, internal ones intersect at the center.
- data: [
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: -2 + 1.5 * Math.cos(angle),
y: 1.5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: Array.from({ length: 361 }, (_, i) => {
const angle = (i * Math.PI) / 180;
return {
x: 2 + 1.5 * Math.cos(angle),
y: 1.5 * Math.sin(angle),
z: 0,
};
}),
color: getColor("PURPLE"),
showPoints: false,
},
{
points: [
{ x: -5, y: 1.5, z: 0 },
{ x: 5, y: 1.5, z: 0 },
],
color: getColor("CYAN"),
showPoints: false,
smooth: false,
},
{
points: [
{ x: -5, y: -1.5, z: 0 },
{ x: 5, y: -1.5, z: 0 },
],
color: getColor("TEAL"),
showPoints: false,
smooth: false,
},
{
points: (() => {
const O1 = { x: -2, y: 0 };
const O2 = { x: 2, y: 0 };
const r1 = 1.5;
const r2 = 1.5;
const d = Math.sqrt((O2.x - O1.x) ** 2 + (O2.y - O1.y) ** 2);

const alpha = Math.asin((r1 + r2) / d);
const theta = Math.atan2(O2.y - O1.y, O2.x - O1.x);

const T1x = O1.x + r1 * Math.cos(theta + Math.PI / 2 - alpha);
const T1y = O1.y + r1 * Math.sin(theta + Math.PI / 2 - alpha);

const T2x = O2.x + r2 * Math.cos(theta + Math.PI / 2 - alpha + Math.PI);
const T2y = O2.y + r2 * Math.sin(theta + Math.PI / 2 - al ... [truncated; 2320 chars]
- cameraPosition: [0, 0, 10]
- showZAxis: false

**Solution:**

Component: MathContainer
Children:

```math
d = \sqrt{(2-(-2))^2 + (0-0)^2} = 4
```

```math
r_1 = r_2 = 1.5
```

```math
l_{external} = d = 4
```

```math
l_{internal} = \sqrt{4^2 - (1.5+1.5)^2} = \sqrt{16 - 9} = \sqrt{7}
```

## Practice Problems

1. Two circles are centered at $$A(-4, 0)$$ with radius $$2$$ and $$B(4, 0)$$ with radius $$3$$. Determine:

   - Length of external common tangent line
   - Length of internal common tangent line

2. The first circle has center $$(0, 0)$$ with radius $$4$$, the second circle has center $$(10, 0)$$ with radius $$2$$. Calculate the length of both types of common tangent lines!

3. Two identical circles each have radius $$3 \text{ cm}$$. If the length of the internal common tangent line is $$8 \text{ cm}$$, determine the distance between the two circle centers!

4. Circle $$A$$ is centered at $$(-5, 0)$$ with radius $$r$$, and circle $$B$$ is centered at $$(7, 0)$$ with radius $$2r$$. If the length of the external common tangent line is $$4\sqrt{8}$$, determine the value of $$r$$!

5. Determine the conditions for two circles to have:
   - Exactly two common tangent lines
   - Exactly three common tangent lines
   - Exactly four common tangent lines

Visible text: 1. Two circles are centered at with radius and with radius . Determine:

 - Length of external common tangent line
 - Length of internal common tangent line

2. The first circle has center with radius , the second circle has center with radius . Calculate the length of both types of common tangent lines!

3. Two identical circles each have radius . If the length of the internal common tangent line is , determine the distance between the two circle centers!

4. Circle is centered at with radius , and circle is centered at with radius . If the length of the external common tangent line is , determine the value of !

5. Determine the conditions for two circles to have:
 - Exactly two common tangent lines
 - Exactly three common tangent lines
 - Exactly four common tangent lines

### Answer Key

1. **Calculating common tangent line lengths**

   <MathContainer>
     
   
   ```math
   d = \sqrt{(4-(-4))^2 + (0-0)^2} = 8
   ```

     
   
   ```math
   l_{external} = \sqrt{8^2 - (3-2)^2} = \sqrt{64 - 1} = \sqrt{63} = 3\sqrt{7}
   ```

     
   
   ```math
   l_{internal} = \sqrt{8^2 - (2+3)^2} = \sqrt{64 - 25} = \sqrt{39}
   ```

   </MathContainer>

2. **Circles with different centers**

   <MathContainer>
     
   
   ```math
   d = \sqrt{(10-0)^2 + (0-0)^2} = 10
   ```

     
   
   ```math
   l_{external} = \sqrt{10^2 - (4-2)^2} = \sqrt{100 - 4} = \sqrt{96} = 4\sqrt{6}
   ```

     
   
   ```math
   l_{internal} = \sqrt{10^2 - (4+2)^2} = \sqrt{100 - 36} = \sqrt{64} = 8
   ```

   </MathContainer>

3. **Finding center distance from internal tangent length**

   Given: $$r_1 = r_2 = 3$$, $$l_{internal} = 8$$

   <MathContainer>
     
   
   ```math
   l = \sqrt{d^2 - (r_1 + r_2)^2}
   ```

     
   
   ```math
   8 = \sqrt{d^2 - 6^2}
   ```

     
   
   ```math
   64 = d^2 - 36
   ```

     
   
   ```math
   d^2 = 100
   ```

     
   
   ```math
   d = 10 \text{ cm}
   ```

   </MathContainer>

4. **Finding the value of** $$r$$

   Given: $$d = 12$$, $$r_1 = r$$, $$r_2 = 2r$$, $$l_{external} = 4\sqrt{8}$$

   <MathContainer>
     
   
   ```math
   4\sqrt{8} = \sqrt{12^2 - (2r - r)^2}
   ```

     
   
   ```math
   16 \cdot 8 = 144 - r^2
   ```

     
   
   ```math
   128 = 144 - r^2
   ```

     
   
   ```math
   r^2 = 16
   ```

     
   
   ```math
   r = 4
   ```

   </MathContainer>

5. **Conditions for number of common tangent lines**

   - **Exactly $$2$$ tangent lines**: The two circles intersect at two points
   - **Exactly $$3$$ tangent lines**: The two circles are tangent (internally or externally)
   - **Exactly $$4$$ tangent lines**: The two circles are separate (do not intersect)

Visible text: 1. **Calculating common tangent line lengths**

 <MathContainer>
 
 

 
 

 
 

 </MathContainer>

2. **Circles with different centers**

 <MathContainer>
 
 

 
 

 
 

 </MathContainer>

3. **Finding center distance from internal tangent length**

 Given: , 

 <MathContainer>
 
 

 
 

 
 

 
 

 
 

 </MathContainer>

4. **Finding the value of** 

 Given: , , , 

 <MathContainer>
 
 

 
 

 
 

 
 

 
 

 </MathContainer>

5. **Conditions for number of common tangent lines**

 - **Exactly tangent lines**: The two circles intersect at two points
 - **Exactly tangent lines**: The two circles are tangent (internally or externally)
 - **Exactly tangent lines**: The two circles are separate (do not intersect)