# Problem in Direct Similarity

### Source

The problem below grew out from an observation by Miguel Ochoa Sanchez:

### Problem

Let $ABC$ and $EBD$ be two similar triangles $M,N,P$ the midpoints of $BE,BC,AD,$ respectively. Define $R,S,T$ as the barycenters of triangles $EBD,MNP,ABC.$

Then $R+T=2S.$

### Proof

We'll see the problem as set in affine geometry.

Let $B = 0,$ the origin, $E = x,$ $D = y,$ $a$ - spiral similarity around $B$ s.t. $A = ax,$ $C = ay.$ Then $M = x/2,$ $N =$ay/2,P = (ax + y)/2.$Further,$R = (x+y)/3,T = (ax+ay)/3,S = (x/2+ay/2+ (ax + y)/2)/3=((x+y)/3+(ax+ay)/3)/2=(R+T)/2.$Note that$\Delta MNP$is not similar to$\Delta ABC.$Using complex numbers, Leo Giugiuc proved that if$ABC$and$EBD$are equilateral then so is$MNP;$and Gregoire Nicollier showed that$MNP$and$EBD$are directly similar if and only if$EBD\$ is equilateral.

### Acknowledgment

The original statement has been posted by Leo Giugiuc at the CutTheKnotMath facebook page with credits to Miguel Ochoa Sanchez.