## If G is the circumcenter of triangle ACE

Distances between the circumcenter **G** and the vertices **A, E**, and **C **of a triangle are equal.

It means,

AG=CG=EG

Substitute the **AG=7x-41, CG=5x-19** into the above expression:

7x-41=5x-19

Collect like terms:

7x-5x=41-19

Simplify The above expression

2x=22

Both sides are divided into **2**:

x=\frac{\cancel{22}}{\cancel{2}}=11

Find value of **AG=7x-41** for x=11:

AG=7\times11-41

After simplifying we get,

AG=77-41=36

We know that from above AG=CG=EG

AG=CG=EG=36

Here, **GD, B**G, and **GF** are perpendicular bisectors of the circumcenter **G**:

triangle **GDE** is a right Triangle whose side **ED=28**:

Let be GD denoted by x:

by using the **Pythagorean Theorem**:

(GD)^2+(ED)^2=EG^2

Substitute the **EG=36, ED=28** into the above

(GD)^2+(28)^2=(36)^2

After simplifying we get

(GD)^2=1296-784=512

The value of GD is:

GD=\sqrt{512}=16\sqrt{2}=22.62

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