In a triangle, the lengths of the two sides are 6 and 3 cm. Find the length of the third side if the half-sum of the heights projected to the given sides is equal to the third height.
My attempt at answering: I drew this as an acute triangle and found that there will be an orthocenter in this triangle. The third side based on my drawing is the base. I just don't understand how to relate and substitute the values given to get the answer. I'm still quite new to this concept of an orthocenter so I don't know much of the formulas or proofs yet. Any links to a helpful website would be much appreciated :)
We have $$\frac{h_a+h_b}{2}=h_c$$ Using that $$A=\frac{1}{2}h_aa=\frac{1}{2}h_bb=\frac{1}{2}h_cc$$ we get $$\frac{1}{a}+\frac{1}{b}=\frac{2}{c}$$ so $$c=\frac{2ab}{a+b}$$