The many ways to construct a triangle

m_a, m_b, m_c

Through A and C draw lines parallel to CM_c and AM_a, repsectively. Let P be the point of intersection. APCG is a parallelogram, so its diagonals are halved by their point of intersection. Therefore they intersect at M_b, and CM_b is a median of GPC. PC is equal and parallel to AG whereas GM_a is 1/2 of AG and, hence, of PC. Therefore, median GK of GPC is parallel to BC and equals a/2. The third median in GPC is similarly parallel to AB and equals c/2.

Then there is the construction. Take 2/3 of each given median. Use SSS to construct the triangle GPC. Find medians in this triangle. Double them. This will give the three sides of the sought triangle. Draw it again using SSS.

M_a, M_b, M_c

The triangle M_aM_bM_c is similar to ABC and is twice as small. Double its sides. Then use SSS to construct ABC.

a, b, m_c

Triangle M_cM_bC has the following sides: a/2, b/2, m_c. It's therefore possible to construct it using SSS. Now extend CM_b to twice its length to obtain A. Next, extend AM_c to twice its length to obtain B.

a, b, m_b

Triangle BM_bC is formed by the following sides: m_b, b/2, a. We may construct it with SSS. Afterwards, extend CM_b to twice its length to obtain A.

H_a, H_b, H_c

A, B, p

Start with drawing a triangle AB'C' with two given angles A and B. However you do this, the triangle you obtain will be similar to the triangle ABC we have to find. On cc, starting with the point A, measure the perimiter of AB'C' (point P) and 2p (point Q). Connect P and C'. Through Q, draw a line parallel to PC' that intersects bb at the point C. From C draw a line parallel to C'B' to obtain B.

A remark is in order here. Two angles completely define a triangle's shape. To determine a triangle uniquely one has to fix its size. Since in similar triangles all linear elements are in the same ratio, any linear element in a triangle uniquely determines that triangle. Therefore, the above solution will work as well in other cases. For example, when we are given A,B, and, say, l_c or H_aM_b.