Denoting the altitudes of any triangle from sides a, b, and c respectively as \(h_a, h_b, \text{ and } h_c\),and denoting the semi-sum of the reciprocals of the altitudes as \(H = (h_a^{-1} + h_b^{-1} + h_c^{-1})/2\) we have

As an example I am trying to give hint based on a question I solved on brilliant recently where altitude lengths were 3,4 and 5 units respectively.If we are given the side lengths a,b,c then easily heron's formula strikes the mind.Here too first find the side lengths in terms of area with help of corresponding altitude by using the formula 'Area of triangle'= (1/2)(base)(corresponding altitude) and then apply Heron's formula.

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Denoting the altitudes of any triangle from sides a, b, and c respectively as \(h_a, h_b, \text{ and } h_c\),and denoting the semi-sum of the reciprocals of the altitudes as \(H = (h_a^{-1} + h_b^{-1} + h_c^{-1})/2\) we have

\[A^{-1} = 4 \sqrt{ H (H - h_a^{-1}) (H-h_b^{-1}) (H-h_c^{-1}) } \]

from Altitude

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Thanks!

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As an example I am trying to give hint based on a question I solved on brilliant recently where altitude lengths were 3,4 and 5 units respectively.If we are given the side lengths a,b,c then easily heron's formula strikes the mind.Here too first find the side lengths in terms of area with help of corresponding altitude by using the formula 'Area of triangle'= (1/2)(base)(corresponding altitude) and then apply Heron's formula.

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