A Flock Of Herons

Geometry Level 5

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

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

Radius of circle A=1092=223713A=1092=2^2\cdot 3\cdot 7\cdot 13
Radius of circle B=608=2519B=608=2^5\cdot 19
Radius of circle C=1140=223519C=1140=2^2\cdot 3\cdot 5\cdot 19

The sides of triangle ΔABC \Delta ABC are

AB=1700=225217AB=1700=2^2\cdot 5^2\cdot 17
BC=5491=17219BC=5491=17^2\cdot 19
CA=4437=321729CA=4437=3^2\cdot 17\cdot 29

Triangle ΔXYZ\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,CA,B,C

Let ΔXYZΔABC=pq\dfrac { \Delta XYZ }{ \Delta ABC } =\dfrac { p }{ q }

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

where p,qp,q are 44 digit co-prime integers. The difference pqp-q is a 44 digit prime number.

Find that prime number.

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

3261654=23311172193261654=2\cdot 3^3\cdot 11\cdot 17^2\cdot 19

Note: Triangle ΔXYZ\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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