# Is it possible to reconstruct a triangle from the midpoints of its sides?

Take $ABC$ an arbitrary triangle, it is easy to take the midpoints $P$, $Q$, $R$ of sides $AB$, $BC$, $CA$, and we all know that the medians $CP$, $AQ$, $BR$ intersect at a point called the centroid (or barycenter) of the triangle.

But, given three points $P$, $Q$, $R$, is it possible, using tools of Euclidean geometry, to reconstruct the triangle (or one of the triangles, if there are many) of which they are the midpoints of sides?

• Yes, and that triangle would be unique! – Emin Feb 20 '14 at 12:51

## 1 Answer

Construct parallelograms from the triangle PQR, taking each side (PQ, QR, RP) to be diagonal of the constructed parallelogram.

For PR diagonal, the fourth vertex of the constructed parallelogram will be A. Similarly for PQ it is B and QR it is C respectively.

• That's exactly what I needed, thank you so much! – yannis Feb 20 '14 at 18:31
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