Drawing triangle

Question

follow the picture:

$m$,$p$ and $Q$ are midpoints of segments we want to draw the triangle and we only have the lengh of $AM$,$BQ$ and $CP$ How to draw the triangle?

Answer 1

In the following mystic hexagon, the lengths of the dotted segments are $2m_a,2m_b,2m_c$ and the point $G$ is the centroid of the dotted triangle:

If the dotted triangle is given, we may construct the mystic hexagon by simply reflecting its centroid with respect to the midpoints of the triangle sides. That magic configuration serves also as a proof of:

Lemma 1. If $a,b,c$ are the side lengths of a triangle, $m_a,m_b,m_c$ are the side lengths of a triangle.

Lemma 2. If $a,b,c$ are the side lenghts of a triangle with area $\Delta$, the area of the triangle with side lengths $m_a,m_b,m_c$ is $\frac{3}{4}\Delta$.

Answer 2

Use the fact that the centroid divides each median into parts in the ratio $2:1$, so you may find it drawing three circles with centres in points $P$, $Q$ and $M$ and radiuses $CP/3$, $BQ/3$ and $AM/3$. The point of their intersection is the centroid of tringle, and then you may draw whole medians and find points $A$, $B$ and $C$.

Answer 3

Easy!

Join PQM. You must see the triangles PQM,APQ,BPM,CQM are congruent.

You have three points P,Q,M

Join to form the triangle.

Draw a circle with centre M and radius AM.

Copy $\angle$ MPQ to form $\angle$ PQA and extend the line. Whereever it meets the circle that is your A.

Done. same for rest of the two