How do I construct an isosceles triangle ABC (|AC| = |BC| = a), given $a-v_a$ and the angle $α$?
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Welcome to math SE. What have you tried? – Alain Remillard Dec 18 '23 at 15:15
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I was thinking of starting with the construction of a smaller triangle where one side is given as $a−v_a$, and an angle of $180-α$. However, I'm not sure how to determine the third element. – cchris Dec 18 '23 at 15:30
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$\upsilon_a$, what does that mean? – Dominique Dec 18 '23 at 15:46
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$v_a$ is altitude to side a – cchris Dec 18 '23 at 15:47
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Could you edit your post to include these elements? – Alain Remillard Dec 18 '23 at 18:11
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The following hint is based on the assumption that you have an isosceles triangle, with the two equal sides of length $~a.~$ Denote one of these sides as $~A.~$ I am then assuming that the angle $~\alpha~$ is opposite $~A~$ and that an altitude $~V_a,~$ runs from angle $~\alpha~$ to side $~A.~$ This implies that one of the angles adjacent to side $~A~$ is $~\beta = (180^\circ - 2\alpha),~$ and that $$\frac{V_a}{a} = \sin(\beta).$$ This allows you to construct an equation $~E = a \times F,~$ where $~E,F~$ are known, with $~E = a - V_a.~$ Thus $~a = \dfrac{E}{F}.$ – user2661923 Dec 18 '23 at 19:05