Suppose we have given a triangle ∆ABC with sides AB=c, BC=a & AC=b, also with O as it's orthocentre and I1 as itsi excentre opposite to angle A. Find the distance between excentre and orthocentre

To find the distance between the excentre \( I_1 \) opposite to angle \( A \) and the orthocentre \( O \) of triangle \( \Delta ABC \), we can use some properties of triangle centers and a specific formula. The orthocentre \( O \) is the point where the altitudes of the triangle intersect, while the excentre \( I_1 \) is the center of the excircle opposite to vertex \( A \). The distance between these two points can be derived using the triangle's sides and angles.

Understanding the Key Concepts

Before diving into the calculations, let’s clarify a few important points:

• Orthocentre (O): The point where the three altitudes of the triangle intersect.
• Excentre (I1): The center of the excircle opposite to vertex \( A \), which is tangent to the extensions of sides \( AB \) and \( AC \).
• Triangle Sides: We denote the sides of the triangle as \( AB = c \), \( BC = a \), and \( AC = b \).

Distance Formula

The distance \( d \) between the orthocentre \( O \) and the excentre \( I_1 \) can be calculated using the following formula:

d = \sqrt{R^2 - 2Rr_1}

Where:

• R: The circumradius of the triangle.
• r_1: The radius of the excircle opposite to angle \( A \).

Calculating R and r_1

To apply this formula, we first need to find \( R \) and \( r_1 \).

Finding the Circumradius (R)

The circumradius \( R \) can be calculated using the formula:

R = \frac{abc}{4K}

Where \( K \) is the area of the triangle, which can be calculated using Heron's formula:

K = \sqrt{s(s-a)(s-b)(s-c)}

Here, \( s \) is the semi-perimeter given by:

s = \frac{a+b+c}{2}

Finding the Exradius (r_1)

The exradius \( r_1 \) opposite to angle \( A \) can be calculated using the formula:

r_1 = \frac{K}{s-a}

Putting It All Together

Once you have calculated \( R \) and \( r_1 \), you can substitute these values into the distance formula:

d = \sqrt{R^2 - 2Rr_1}

This will give you the distance between the orthocentre \( O \) and the excentre \( I_1 \). It’s important to ensure that all calculations are done accurately, especially when dealing with the area and the semi-perimeter.

Example Calculation

Let’s consider a specific triangle with sides \( a = 7 \), \( b = 8 \), and \( c = 9 \). First, we calculate the semi-perimeter:

s = \frac{7 + 8 + 9}{2} = 12

Next, we calculate the area \( K \) using Heron’s formula:

K = \sqrt{12(12-7)(12-8)(12-9)} = \sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} = 12\sqrt{5}

Now, we can find \( R \):

R = \frac{7 \times 8 \times 9}{4 \times 12\sqrt{5}} = \frac{504}{48\sqrt{5}} = \frac{21}{2\sqrt{5}}

Next, we calculate \( r_1 \):

r_1 = \frac{12\sqrt{5}}{12-7} = \frac{12\sqrt{5}}{5}

Finally, substitute \( R \) and \( r_1 \) into the distance formula to find \( d \).

Final Thoughts

This process illustrates how to derive the distance between the orthocentre and the excentre in a triangle. By understanding the relationships between these triangle centers and applying the relevant formulas, you can solve similar problems effectively. If you have any further questions or need clarification on any steps, feel free to ask!