Editing Altitude (triangle) (section)

===Special cases===

====Equilateral triangle====

From any point {{mvar|P}} within an [[equilateral triangle]], the sum of the perpendiculars to the three sides is equal to the altitude of the triangle.  This is [[Viviani's theorem]].

====Right triangle====
{{right_angle_altitude.svg}}
[[File:inverse_pythagorean_theorem.svg|thumb|Comparison of the inverse Pythagorean theorem with the Pythagorean theorem]]
In a right triangle with legs {{mvar|a}} and {{mvar|b}} and hypotenuse {{mvar|c}}, each of the legs is also an altitude: {{tmath|1= h_a = b}} and {{tmath|1= h_b = a}}. The third altitude can be found by the relation<ref>Voles, Roger, "Integer solutions of <math>a^{-2}+b^{-2}=d^{-2}</math>," ''Mathematical Gazette'' 83, July 1999, 269–271.</ref><ref>Richinick, Jennifer, "The upside-down Pythagorean Theorem," ''Mathematical Gazette'' 92, July 2008, 313–317.</ref>

:<math>\frac{1}{h_c ^2} = \frac{1}{h_a ^2}+\frac{1}{h_b ^2} = \frac{1}{a^2}+\frac{1}{b^2}.</math>

This is also known as the [[inverse Pythagorean theorem]].

Note in particular:
:<math>\begin{align}
\tfrac{1}{2} AC \cdot BC &= \tfrac{1}{2} AB \cdot CD \\[4pt]
         CD &= \tfrac{AC \cdot BC}{AB} \\[4pt]
\end{align}</math>