# Euclidean Geometry Construction

I am looking for an aswer to the following construction construct a triangle given two angles (3 angles) and the sum of two sides

• There are (except for some special angles) three such triangles. They can all be constructed fairly easily by straightedge and compass. Just construct a triangle with correct angles, and scale. – André Nicolas Oct 16 '14 at 18:02
• Only the sum of two sides is given ! – Philippe Sussholz Oct 16 '14 at 18:12

Step 1: Draw a line $AB$ of arbitrary length. Copy angle $\alpha$ at point $A$ and angle $\beta$ at point $B$ to form $\triangle ABC$.

Step 2: Extend line $AB$ in the direction of $B$ by length $AC$, calling that line $AD$.

Step 3: Draw line a perpendicular to $AD$ at $A$, and mark off the given required sum of two sides $s$ such that the length of $AE$ is $s$. Connect points $D$ and $E$ to form line $DE$.

Step 4: construct a line parallel to $DE$ passing through point $B$. This line intersects line $AE$ at some point $F$.

Step 5: Copy angle $\alpha$ such that one leg is on $AF$ and the vertex is at $A$. Copy angle $\alpha$ such that one leg is on $AF$ and the vertex is at $A$. Copy angle $\beta$ such that one leg is on $AF$ and the vertex is at $F$. The two new lines meet at some point $G$.

Triangle $AFG$ is the required triangle.

Note that the solution is not unique, unless the problem has specified that the sum of two sides meeting at a specific one of the angles must be the given sum.

• I am sorry but this answer doesn't work. It seems that AF is omger than AE which is the orignal a+b. – Philippe Sussholz Oct 16 '14 at 19:07
• I am going to rephrase my question . I have a triangle with the three angles , and one side is a+b . – Philippe Sussholz Oct 16 '14 at 19:09
• I need to find a SIMILAR triangle with sides a and b – Philippe Sussholz Oct 16 '14 at 19:10