# A Flock Of Herons

Geometry Level 5

Triangles $$\Delta ABC$$ and $$\Delta XYZ$$ are Heronian triangles, which are triangles that have integer sides with integer areas.

Points $$A,B,C$$ are centers of circles, the radii of which are

Radius of circle $$A=1092=2^2\cdot 3\cdot 7\cdot 13$$
Radius of circle $$B=608=2^5\cdot 19$$
Radius of circle $$C=1140=2^2\cdot 3\cdot 5\cdot 19$$

The sides of triangle $$\Delta ABC$$ are

$$AB=1700=2^2\cdot 5^2\cdot 17$$
$$BC=5491=17^2\cdot 19$$
$$CA=4437=3^2\cdot 17\cdot 29$$

Triangle $$\Delta XYZ$$ has the maximum area of any triangle (Heronian or not) that has one vertex on the circumference of each of the circles $$A,B,C$$

Let $$\dfrac { \Delta XYZ }{ \Delta ABC } =\dfrac { p }{ q }$$

be the ratio of the areas of triangles $$\Delta XYZ$$ and $$\displaystyle \Delta ABC$$

where $$p,q$$ are $$4$$ digit co-prime integers. The difference $$p-q$$ is a $$4$$ digit prime number.

Find that prime number.

For your convenience, area of triangle $$\displaystyle \Delta ABC$$ works out to

$$3261654=2\cdot 3^3\cdot 11\cdot 17^2\cdot 19$$

Note: Triangle $$\Delta XYZ$$ as drawn in graphic does not have the maximum area

Also, as you solve this, you'll come across more Herons. A helpful hint, I hope.

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