Extra geometry III

Let D be a point inside an acute angled triangle ABC. Show that:

(DA)(DB)(AB) + (DB)(DC)(BC) + (DC)(DA)(CA) &ge (AB)(BC)(CA)

with equality if and only if D is the orthocentre of ABC.

Extra geometry II

Let ABC be a triangle with the angle at A equal to 40 degrees and that at B equal to 60 degrees. Let D and E be the points lying on AC and AB respectively so that angle CBD = 40 degrees and angle BCE = 70 degrees. Let BD and CE meet at F. Show that the line AF is perpendicular to BC.

Extra geometry I

A convex quadrilateral ABCD has AD = CD and angles DAB and ABC are equal and acute. The line through D and the midpoint of BC intersects AB at E. Prove that angles BEC and DAC are equal.

Geometry

Our next assignment (coming soon I promise) will be on geometry. So you might want to spend some time doing a bit of reading of whatever materials you have available.

At the EMAT 4600/6600 webpage towards the bottom you'll find some (fairly elementary) problems with hints and solutions as well as some review of many of the standard theorems. That would be a useful place to start (though Google "geometry problems" will also provide you with much more than you might want to look at -- if you do find something interesting with that, please let me know!)

Have fun.