P is a point inside an equilateral triangle whose altitude measures 15 cm. The perpendicular distances from P to two of the three sides are 4 cm and 6 cm. What is the perpendicular distance from P to the third side?

Join P to the three vertices. The three small triangles have the three sides of the equilateral triangle as bases — all equal, say s — and the three perpendicular distances as heights, and together they make up the whole triangle. So ½s(d₁ + d₂ + d₃) = ½s × h, which forces d₁ + d₂ + d₃ = h whatever the position of P (Viviani's theorem). Hence the third distance = 15 − 4 − 6 = 5 cm. Subtracting only one of the two given distances gives the trap 10 cm, and 7.5 cm just halves the altitude.