Take any point P in the interior or boundary of △ ABC, and note that the distances from P to three sides a, B, C are x, y, Z in turn. Prove that ax + by + CZ is a constant

Take any point P in the interior or boundary of △ ABC, and note that the distances from P to three sides a, B, C are x, y, Z in turn. Prove that ax + by + CZ is a constant

Prove: as shown in the figure, connect PA, Pb, PC, divide △ ABC into three small triangles, then s △ ABC = s △ PAB + s △ PCB + s △ PCA = 12CZ & nbsp; + 12ax + 12by, so ax + by + CZ = 2S △ ABC, that is, ax + by + CZ is a constant. Explain that if △ ABC is an equilateral triangle, then x + y + Z = 2S △ ABCA = h, that is, the sum of distances from one point to three sides of an equilateral triangle is a constant, and this constant is the height of an equilateral triangle