Rational height of a Triangle inside a Triangle

Geometry Level 4

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Let ABC be a given triangle with AB = AC = 5 and BC = 3, a point P1 is on BC such that BP1 = 1.

Points P2 and P3 are chosen on sides AB, AC respectively such that they are at the shortest distance from P1.

Find the height of the triangle P1P2P3, with respect to the base P2, P3.

While I was not interested in a irrational solution, the best i could do was to express the height as follows

\[ h = \frac{ a*b} { (c^d) *d * \sqrt{e} } \]

where a, b, c, d, e are distinct prime integers,

find a + b + c + d + e ?

Source: Own work.


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