Viviani's Theorem

For a point $P$ inside an Equilateral Triangle $\Delta ABC$, the sum of the perpendiculars $p_i$ from $P$ to the sides of the Triangle is equal to the Altitude $h$. This result is simply proved as follows,

$$\Delta ABC=\Delta PBC+\Delta PCA+\Delta PAB.$$ (1)

With $s$ the side length,

$${\textstyle{1\over 2}}sh={\textstyle{1\over 2}}sp_a+{\textstyle{1\over 2}}sp_b+{\textstyle{1\over 2}}sp_c,$$ (2)

so

$$h=p_a+p_b+p_c.$$ (3)

See also Altitude, Equilateral Triangle