Distance of The Orthocenter from Vertices and Sides of a Triangle
Distance of The Orthocenter from Vertices and Sides of a Triangle:
\(In\ \Delta ADB,\ \angle BAD=\frac{\pi }{2}-B\),
\(AF=b\cos A\),
( Projection of AC on AB)
In triangle AFH, \(\cos \left( \frac{\pi }{2}-B \right)=\frac{AF}{AH}\),
\(=\frac{b\cos A}{AH}\),
\(\Rightarrow AH=\frac{b\cos A}{\cos \left( \frac{\pi }{2}-B \right)}\),
= \(\frac{b\cos A}{\sin B}\),
\(=2R\cos A\)Similarly, BH = 2RcosB and CH = 2R cosC
\(In\ \ \Delta AFH,\ \tan \left( \frac{\pi }{2}-B \right)\) =FH/AF = FH/b cosA
cotB = FH/ b cosA
FH = b cosA cotB
FH = (bcosA cosB)/sinB
=2R cosAcosB
Similarly, EH = 2R cosA cosC
And HD = 2RcosBcosC