To solve this problem without using Heron's formula, we can make use of properties of a triangle inscribed around a circle (an incircle). Here's the step-by-step solution:

Step 1: Understanding the Given Data and Setting Up Variables

• We have a triangle $ABC$ circumscribing a circle with radius $r = 4$ cm.
• The incircle touches side $BC$ at point $D$, dividing $BC$ into segments $BD = 8$ cm and $DC = 6$ cm.

Let:

• $AB = c$
• $AC = b$
• $BC = a$

Thus, $a = BD + DC = 8 + 6 = 14$ cm.

Step 2: Apply the Incircle Property

The incircle divides each side into two segments such that each segment is equal to the semiperimeter ($s$) minus the length of the opposite side.

So we know:

• $BD = s - c$
• $DC = s - b$

Step 3: Calculate the Semiperimeter $s$

The semiperimeter $s$ of the triangle is given by: $s = \frac{a + b + c}{2}$

Since $BD = 8$ and $DC = 6$, we have: $s - c = 8 \quad \text{and} \quad s - b = 6$

Step 4: Express $b$ and $c$ in Terms of $s$

Rearrange the equations to find $b$ and $c$ in terms of $s$: $c = s - 8$ $b = s - 6$

Step 5: Use the Area Formula for an Incircle

The area $A$ of a triangle with an incircle is given by: $A = s \cdot r$ Since $r = 4$ cm, we have: $A = 4s$

Step 6: Calculate the Area in Terms of $a$, $b$, and $c$

The area $A$ can also be expressed using the base $a$ and the altitude (height) to that base, which we calculate via: $A = \frac{1}{2} \times a \times h$ where $h$ is the height from $A$ to $BC$.

Equating both expressions for the area $A$: $4s = \frac{1}{2} \times a \times h$

Since we are not given $h$, we proceed with finding the values for $s$, $b$, and $c$ directly by substituting values and solving accordingly.