Sankhadip Chakraborty
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Such numbers are known as repunits. A repunit with (n) digits is denoted by (R_n). It is known that (R_n) is a prime for (n=2,19,23,317)and 1031. (Source : Wikipedia).
Hi,
Note that the nine-point centre lies halfway between the circumcentre and the orthocentre. If both of them lie inside the triangle, this cannot happen. Hence, at least one of them has to lie outside. This means that the triangle has to be obtuse (a right-angled triangle does not work because in that case, the orthocentre coincides with a vertex). Hence, the image you should have in mind is that of an obtuse triangle with the obtuse angle being A.
Take any point \(Q\) on \(b\). Through \(Q\), draw a line \(a'\) which makes the given angle with \(b\). Now draw a parallel to \(a'\) through \(P\). This is our required \(a\).
Let us denote \(M=XO\cap QR\). It is clear that the triangle \(OMQ\) is right isosceles and \(OQ=2\). From this information, we can find \(OM\). This in turn gives the area of the bounded arc \(QRO\). Again, we know that \(\angle QOR=\frac{\pi}{2}\) hence we know the area of the sector bounded by the arc \(QXR\). Subtracting the area of the triangle \(QRO\) we get the area of the region bounded by \(QR\) and the arc \(QXR\). Convince yourself that this is all we need.