Challenge Level

Clara from Herlufsholm School in Denmark sent in some observations:

I noticed that the quadrilaterals formed by joining the midpoints always has two pairs of parrallel sides. This means that the shape formed is always a square, rectangle or parallelogram. There didn't seem to be any counter-examples to this. This is because when you move one point (For example C) and keep all the other points the same, two points of the new shape move (R and S). This means that the top and bottom edges do not change angle (RS and QP). The other two sides however move but with the same amount and therefore also stay parallel.

PQRS always has half of the area of ABCD. One way to show that this is always true could be to draw a line through the points SQ and RP as in the picture attached. When you take a quarter of the whole image there is another quadrilateral with a line through the middle (as shown in the picture underneath figure 1).

If figure ABCD is concave, the ration of the area of ABCD and PQRS is still 2:1.

Clara's observations and predictions are correct, but this way of showing that PQRS has half the area of ABCD won't always work. For example, in Clara's picture 3, the triangles in the pieces are not the same size. This is shown below. In the image on the right, the yellow triangle has been rotated to show that it has a smaller area than the red triangle.

Samvit from Mason Middle School in the USA sent this rigorous proof:

The new quadrilateral will always be a parallelogram.

For this proof, CASTC means Corresponding Angles of Similar Triangles are Congruent, which is part of the definition of a similar triangle (like CPCTC).

Take this arbitrary image of a quadrilateral, where

E is the midpoint of segment AB.

F is the midpoint of segment BC.

G is the midpoint of segment CD.

H is the midpoint of segment DA.

Two column proof:

Statement | Reason |
---|---|

E is the midpoint of segment AB | Given |

segment AE is congruent to segment EB | Definition of a midpoint |

AE = EB | Definition of congruence |

AE + EB = AB | segment addition postulate |

EB + EB = AB | Substitution |

2(EB) = AB | Simplify |

angle EBF is congruent to angle ABC | Reflexive property |

F is the midpoint of BC | Given |

segment FC is congruent to segment BF | Definition of a midpoint |

FC = BF | Definition of congruence |

FC + BF = BC | Segment addition postulate |

BF + BF = BC | Substitution |

2(BF) = BC | Simplify |

triangle EBF is similar to triangle ABC | SAS similarity |

angle CAB is congruent to angle FEB | CASTC |

Segment AC is parallel to segment EF | Converse of Corresponding Angles |

G is the midpoint of segment CD | Given |

Segment GC is congruent to segment GD | Definition of a midpoint |

GC = GD | Definition of congruence |

GC + GD = CD | segment addition postulate |

GD + GD = CD | Substitution |

2(GD) = CD | Simplify |

angle CDA is congruent to angle GDH | Reflexive property |

H is the midpoint of DA | Given |

segment AH is congruent to segment DH | Definition of a midpoint |

AH = DH | Definition of congruence |

AH + DH = DA | Segment addition postulate |

DH + DH = DA | Substitution |

2(DH) = DA | Simplify |

triangle CDA is similar to triangle GDH | SAS similarity |

angle DAC is congruent to angle DHG | CASTC |

Segment AC is parallel to segment HG | Converse of Corresponding Angles |

Segment EF is parallel to segment HG | Transitive property of parallel lines |

AE + EB = AB | Segment Addition postulate |

AE + AE = AB | substitution |

2(AE) = AB | Simplify |

Angle HAE is congruent to angle DAB | Reflexive property |

AH + DH = DA | Segment Addition postulate |

AH + AH = DA | Substitution |

2(AH) = DA | Simplify |

Triangle HAE is similar to triangle DAB | SAS similarity |

Angle AHE is congruent to angle ADB | CASTC |

Segment DB is parallel to segment HE | Converse of Corresponding Angles |

FC + BF = BC | Segment Addition Postulate |

FC + FC = BC | Substitution |

2(FC) = BC | Simplify |

Angle FCG is congruent to angle BCD | Reflexive |

GC + GD = CD | Segment Addition Postulate |

GC + GC = CD | Substitution |

2(GC) = CD | Simplify |

Triangle FCG is similar to triangle BCD | SAS Similarity |

Angle CGF is congruent to angle CDB | CASTC |

Segment DB is parallel to segment GF | Converse of Corresponding Angles |

Segment HE is parallel to segment GF | Transitive property of parallel lines |

EFGH is a parallelogram | Definition of a parallelogram (opposite sides are parallel) |

Marmik from India proved that the area of PQRS is half of the area of ABCD using the formula for the area of a triangle, $\frac12ab\sin{C}$. Click here to see Marmik's work.

Ana used coordinate geometry to derive the two results. Click here to see Ana's work.