Let PQR be a triangle. Take a point A on or inside the triangle. Let f(x, y) = ax + by + c. Show that
Basic idea is this: First we take A on a side, say PQ. We show
Next we take A in the interior. Lets join RA and produce it to meet PQ at T. Using the previous argument we show
Hence it is sufficient to show for a point A on the line PQ (and the rest will follow):
Let Since A is on PQ, it is possible to write
Suppose f(A) is larger than both f(P) and f(Q). Then f(A) - f(P) and f(A) - f(Q) are both positive.
Similarly we can show
Hence f(A) - f(Q) and f(A) - f(P) are opposite signs. The conclusion follows.