Triangle ABC with angles alpha, beta, gamma respectively and the points P, Q and R marked as the points opposite A, B and C respectively, in the sense that the line which joins them is perpendicular to the opposite side

A triangle \(ABC\) has sides \(BC\), \(CA\), and \(AB\) of lengths \(a\), \(b\) and \(c\) respectively, and angles at \(A\), \(B\) and \(C\) are \(\alpha\), \(\beta\) and \(\gamma\) where \(0 \leq \alpha ,\beta ,\gamma \leq \frac{1}{2}\pi\).

  1. Show that the area of \(ABC\) equals \(\frac{1}{2}bc\sin\alpha\).

    Deduce the sine rule \[\frac{a}{\sin\alpha} = \frac{b}{\sin\beta} = \frac{c}{\sin\gamma}.\]

  2. The points \(P\), \(Q\) and \(R\) are respectively the feet of the perpendiculars from \(A\) to \(BC\), \(B\) to \(CA\), and \(C\) to \(AB\) as shown.

    Prove that \[\text{Area of $PQR$} = (1 - \cos^2 \alpha - \cos^2 \beta - \cos^2 \gamma) \times (\text{Area of $ABC$}).\]

  3. For what triangles \(ABC\), with angles \(\alpha\), \(\beta\), \(\gamma\) as above, does the equation \[\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma =1\] hold?