The objective is to determine the value $q$ for a specific angle $γ$.

Given is a triangle with the known values:

  • base length $c$,
  • height $hc$ relative to this base,
  • the angle $γ$ opposite this base.

The challenge arises due to the ambiguity created by the triangle inequality and the fact that, for a given value of $γ$, there are two possible configurations of the triangle. This results in two potential values for $q$: $q$ and $-q$

Approach So Far: Various trigonometric relationships and triangle properties have been utilized to express $q$, but it has not been possible to derive a closed form for $q$ solely in terms of $c$, $hc$, and $γ$. With more given Values it is possible to calculate $q$ with a specific angle $γ$:

$$q = \tan(90°-α) ⋅ a ⋅ \sin(α-γ)$$

but this way, I can only calculate the given triangle with angle $α$ and $a$


The two possible configurations:

two possible configurations with γ


1 Answer 1


Let's start with the area: $A=\frac{1}{2}c \cdot h=\frac{1}{2}ab \cdot \sin \gamma \implies b=\frac{ch}{a \sin \gamma}$

Next, apply law of cosine: $c^2=a^2+b^2-2ab\cos \gamma$, substituting $b$ we get the equation to find $a^2$: $c^2=a^2 + \frac{c^2h^2}{a^2\sin ^2 \gamma}-2ch \cot \gamma$

This is a quadratic equation in terms of $a^2$, so we can express $a^2:$ $$a^2=0.5 \cdot \left(2ch\cot \gamma+c^2+\sqrt{(2ch\cot \gamma+c^2)^2-\frac{4c^2h^2}{\sin ^2 \gamma}}\right)$$ Note that the problem is symmetrical relative to $a$ and $b$ so the solution will cover both possible configurations. Also, if the expression under the square root is negative, there is no solution which takes care of the triangle inequality.
Finally, $q^2=a^2-h^2$


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