# Constructing triangle $\triangle ABC$ given median to the side $c$ and angles $\alpha$ and $\beta$

Constructing triangle $$\triangle ABC$$ given median to the side $$c$$ and angles $$\alpha$$ and $$\beta$$

I started with the median. Then I constructed a circle to each side of the median, such that the circle is a set of all points which are the vertices of angle $$\alpha$$ and $$\beta$$ above the given segment (the median) respectively. Now, if I only could create a line segment $$AB$$, such that it passes the point $$C_1$$ (the endpoint of the median and the middle point of side $$AB$$), each endpoint is on a different circle (to get the correct angles), and the line segments $$AC_1$$ and $$C_1B$$ are the same length ($$C_1$$ is the endpoint of a median), I would be done. Any ideas?

• Consider the midpoint of the centers of the circles.
– Blue
Nov 26 '20 at 9:28
• Got it, thanks!
– Jeff
Nov 26 '20 at 10:20

So, based on the suggestion of @Blue, the point is to find the midpoint of the centers of the circles $$S$$, construct a line segment $$SC_1$$, and then a line perpendicular to $$SC_1$$ passing through $$C_1$$. The two intersections with the circles are our points $$A$$ and $$B$$.
You can also employ this approach: draw a triangle $$ABC$$ with $$\widehat{A}=\alpha$$ and $$\widehat{B}=\beta$$, consider the length of the median $$MA$$ and apply a homotethy to $$ABC$$ in such a way that the length of $$MA$$ becomes the wanted length.