# Problem in Direct Similarity

### What Might This Be About?

### 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.

### Barycenter and Barycentric Coordinates

- 3D Quadrilateral - a Coffin Problem
- Barycentric Coordinates
- Barycentric Coordinates: a Tool
- Barycentric Coordinates and Geometric Probability
- Ceva's Theorem
- Determinants, Area, and Barycentric Coordinates
- Maxwell Theorem via the Center of Gravity
- Bimedians in a Quadrilateral
- Simultaneous Generalization of the Theorems of Ceva and Menelaus
- Three glasses puzzle
- Van Obel Theorem and Barycentric Coordinates
- 1961 IMO, Problem 4. An exercise in barycentric coordinates
- Centroids in Polygon
- Center of Gravity and Motion of Material Points
- Isotomic Reciprocity
- An Affine Property of Barycenter
- Problem in Direct Similarity
- Circles in Barycentric Coordinates
- Barycenter of Cevian Triangle
- Concurrent Chords in a Circle, Equally Inclined

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