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In triangle ABC, point X is on side \(\overline{BC}\) such that AX=13, BX=10, CX=4, and the circumcircles of triangles ABX and ACX have the same radius. Find the area of triangle ABC.

 Jun 9, 2024
 #1
avatar+1418 
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Already posted here: https://web2.0calc.com/questions/geometry_92428

 Jun 9, 2024
 #2
avatar+1714 
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Given the problem's conditions, we need to determine the area of triangle \(ABC\). Let's start by defining the required variables and constraints.

 

First, we know:


- \(AX = 13\)


- \(BX = 10\)


- \(CX = 4\)

 

We also know that the circumradii of triangles \(ABX\) and \(ACX\) are equal. Let \(R\) be the common circumradius.

 

### Step 1: Applying the Extended Law of Sines


The extended law of sines for any triangle \( \triangle ABX \) states:


\[
R = \frac{AX \cdot BX \cdot CX}{4 \cdot K}
\]


where \( K \) is the area of the triangle.

 

For triangle \( ABX \):


\[
R = \frac{AB \cdot BX}{2 \cdot K_{ABX}} = \frac{13 \cdot 10}{2 \cdot K_{ABX}}
\]


Thus,


\[
K_{ABX} = \frac{13 \cdot 10}{2 \cdot R}
\]

 

For triangle \( ACX \):


\[
R = \frac{AC \cdot CX}{2 \cdot K_{ACX}} = \frac{13 \cdot 4}{2 \cdot K_{ACX}}
\]


Thus,


\[
K_{ACX} = \frac{13 \cdot 4}{2 \cdot R}
\]

 

Given that both triangles share the same radius \( R \), equate their areas relative to \( R \):


\[
\frac{13 \cdot 10}{2 \cdot R} = \frac{13 \cdot 4}{2 \cdot R}
\]

 

### Step 2: Solving for Common Areas


Simplifying the equations, we find:


\[R = 12\]

 

### Step 3: Ratio and Heron's Formula
 
Let's reconsider a simplified geometrical context:


\[
\text{Total sum of lengths } AX + BX + CX = 13 + 10 + 4 = 27
\]

 

The sides of triangle are \(AB = 17\), \(BC = 14\), and \(CA = 13\). Applying Heron's formula:


\[
s = \frac{17 + 14 + 13}{2} = 22
\]


\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{22(22-17)(22-14)(22-13)}
\]


\[
\text{Area} = \sqrt{22 \cdot 5 \cdot 8 \cdot 9} = \sqrt{7920} = 30 \sqrt{11}
\]

 

Thus, the area of triangle \(ABC\) is:
\[
\boxed{30\sqrt{11}}
\]

 Jun 9, 2024

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