Suppose you a have triangle ABC of sides 15, 13 and 4. Three parabolae are drawn such that each parabola is tangent to two side of the triangle such that the third side is a chord of the parabola, which means the two tangents touch the parabola at the vertices of the triangle. Find the area common to the 3 parabolae.
I am a high school student and my maths teacher just asked this question in class. He then told us to not waste our time on it as it is way beyond our level but this keeps bugging me. I have only studied coordinate geometry(algebraic) so I have very little experience in pure geometry of parabola.
I tried to find some symmetry in the question like 5,12,13 and 9,12,15 are pythagorean triplets. I know of a formula which describes the locus of a point from which tangents subtend a particular angle to the parabola which I tried applying as I know all of the angels of the triangle but it was very tedious. I thought maybe the chords are focal chords but I realised tangents at ends of focal chord subtend right angles, so that's also not possible. In the end none of attempts yielded anything fruitful.
So how to solve this problem? Is there an elegant solution without any messy algebraic calculations? Bonus points for book suggestions having these type of tough problems within the high school curriculum.