I asked Y13 student and geometry aficionado Adem K for a WMC problem and this was his choice - it's a very neat configuration of points in a triangle that can lead to other interesting properties!

This gets difficult in places, but everything can be done using GCSE circle theorems and basic angle reasoning. Make sure to draw a good (and big) diagram.

The setup:

• Let ABC be an acute-angled triangle. (The result holds for other triangles, but the diagram gets messy.)
• Draw the altitude from A to BC (this is the line from A that meets BC at a right angle). Let D be the point where the altitude meets BC.
• Similarly, draw the altitude from B to AC, and let E be the intersection with AC. Draw the altitude from C to AB, and let F be the intersection with AB.
• You may assume without proof that the three altitudes meet at a common point H - this is called the orthocentre of the triangle.
• Finally, let L be the midpoint of BC, M be the midpoint of AC, and N be the midpoint of AB.

Now onto the problem!

1. Show that BCEF is a cyclic quadrilateral, and that L is the centre of its circle.
2. Show that DEFL is a cyclic quadrilateral. Deduce that DNEFML is a cyclic hexagon.
3. Show that the midpoints of AH, BH and CH also lie on this circle.

This circle - which passes through D, E, F, L, M, N and the midpoints of AH, BH and CH is called the "nine-point circle". Besides being a nice configuration in its own right, it leads to some interesting results - possibly to appear in a WMC one day!